how to solve this logarithm problem

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How to Solve Logarithms

Last Updated: March 17, 2024 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 262,352 times.

Logarithms might be intimidating, but solving a logarithm is much simpler once you realize that logarithms are just another way to write out exponential equations. Once you rewrite the logarithm into a more familiar form, you should be able to solve it as you would solve any standard exponential equation.

Before You Begin: Learn to Express a Logarithmic Equation Exponentially [1] X Research source [2] X Research source

• If and only if: b y = x
• b does not equal 1
• In the same equation, y is the exponent and x is the exponential expression that the logarithm is set equal to.

• Example: 1024 = ?

• This could also be written as: 4 5

• Example: 4 5 = 1024

Method One: Solve for X

• log 3 ( x + 5) + 6 - 6 = 10 - 6
• log 3 ( x + 5) = 4

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 4; b = 3; x = x + 5
• Rewrite the equation so that: b y = x
• 3 4 = x + 5

• 3 * 3 * 3 * 3 = x + 5
• 81 - 5 = x + 5 - 5

• Example: x = 76

Method Two: Solve for X Using the Logarithmic Product Rule [3] X Research source [4] X Research source

• log b (m * n) = log b (m) + log b (n)

• log 4 (x + 6) + log 4 (x) = 2 - log 4 (x) + log 4 (x)
• log 4 (x + 6) + log 4 (x) = 2

• log 4 [(x + 6) * x] = 2
• log 4 (x 2 + 6x) = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 4 ; x = x 2 + 6x
• 4 2 = x 2 + 6x

• 4 * 4 = x 2 + 6x
• 16 = x 2 + 6x
• 16 - 16 = x 2 + 6x - 16
• 0 = x 2 + 6x - 16
• 0 = (x - 2) * (x + 8)
• x = 2; x = -8

• Example: x = 2
• Note that you cannot have a negative solution for a logarithm, so you can discard x - 8 as a solution.

Method Three: Solve for X Using the Logarithmic Quotient Rule [5] X Research source

• log b (m / n) = log b (m) - log b (n)

• log 3 (x + 6) - log 3 (x - 2) = 2 + log 3 (x - 2) - log 3 (x - 2)
• log 3 (x + 6) - log 3 (x - 2) = 2

• log 3 [(x + 6) / (x - 2)] = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2)
• 3 2 = (x + 6) / (x - 2)

• 3 * 3 = (x + 6) / (x - 2)
• 9 = (x + 6) / (x - 2)
• 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2)
• 9x - 18 = x + 6
• 9x - x - 18 + 18 = x - x + 6 + 18
• 8x / 8 = 24 / 8

• Example: x = 3

Community Q&A

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• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut43_logfun.htm#logdef
• ↑ https://www.mathsisfun.com/algebra/logarithms.html
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut46_logeq.htm
• ↑ https://www.youtube.com/watch?v=fnhFneOz6n8
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut44_logprop.htm

About This Article

To solve a logarithm, start by identifying the base, which is "b" in the equation, the exponent, which is "y," and the exponential expression, which is "x." Then, move the exponential expression to one side of the equation, and apply the exponent to the base by multiplying the base by itself the number of times indicated in the exponent. Finally, rewrite your final answer as an exponential expression. To learn how to solve for "x" in a logarithm, scroll down! Did this summary help you? Yes No

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• What is logarithm equation?
• A logarithmic equation is an equation that involves the logarithm of an expression containing a varaible.
• What are the 3 types of logarithms?
• The three types of logarithms are common logarithms (base 10), natural logarithms (base e), and logarithms with an arbitrary base.
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• When there's no base on the log it means the common logarithm which is log base 10.
• What is the inverse of log in math?
• The inverse of a log function is an exponantial.

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6.3 Logarithmic Functions

Learning objectives.

In this section, you will:

• Convert from logarithmic to exponential form.
• Convert from exponential to logarithmic form.
• Evaluate logarithms.
• Use common logarithms.
• Use natural logarithms.

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes 4 . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, 5 like those shown in Figure 1 . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale 6 whereas the Japanese earthquake registered a 9.0. 7

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is 10 8 − 4 = 10 4 = 10,000 10 8 − 4 = 10 4 = 10,000 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is 10 x = 500 , 10 x = 500 , where x x represents the difference in magnitudes on the Richter Scale . How would we solve for x ? x ?

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve 10 x = 500. 10 x = 500. We know that 10 2 = 100 10 2 = 100 and 10 3 = 1000 , 10 3 = 1000 , so it is clear that x x must be some value between 2 and 3, since y = 10 x y = 10 x is increasing. We can examine a graph, as in Figure 2 , to better estimate the solution.

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function y = b x y = b x is one-to-one , so its inverse, x = b y x = b y is also a function. As is the case with all inverse functions, we simply interchange x x and y y and solve for y y to find the inverse function. To represent y y as a function of x , x , we use a logarithmic function of the form y = log b ( x ) . y = log b ( x ) . The base b b logarithm of a number is the exponent by which we must raise b b to get that number.

We read a logarithmic expression as, “The logarithm with base b b of x x is equal to y , y , ” or, simplified, “log base b b of x x is y . y . ” We can also say, “ b b raised to the power of y y is x , x , ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since 2 5 = 32 , 2 5 = 32 , we can write log 2 32 = 5. log 2 32 = 5. We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

Note that the base b b is always positive.

Because logarithm is a function, it is most correctly written as log b ( x ) , log b ( x ) , using parentheses to denote function evaluation, just as we would with f ( x ) . f ( x ) . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as log b x . log b x . Note that many calculators require parentheses around the x . x .

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = log b ( x ) y = log b ( x ) and y = b x y = b x are inverse functions.

Definition of the Logarithmic Function

A logarithm base b b of a positive number x x satisfies the following definition.

For x > 0 , b > 0 , b ≠ 1 , x > 0 , b > 0 , b ≠ 1 ,

• we read log b ( x ) log b ( x ) as, “the logarithm with base b b of x x ” or the “log base b b of x . " x . "
• the logarithm y y is the exponent to which b b must be raised to get x . x .

Also, since the logarithmic and exponential functions switch the x x and y y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

• the domain of the logarithm function with base b   is   ( 0 , ∞ ) . b   is   ( 0 , ∞ ) .
• the range of the logarithm function with base b   is   ( − ∞ , ∞ ) . b   is   ( − ∞ , ∞ ) .

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

Given an equation in logarithmic form log b ( x ) = y , log b ( x ) = y , convert it to exponential form.

• Examine the equation y = log b ( x ) y = log b ( x ) and identify b , y , and x . b , y , and x .
• Rewrite log b ( x ) = y log b ( x ) = y as b y = x . b y = x .

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

• ⓐ log 6 ( 6 ) = 1 2 log 6 ( 6 ) = 1 2
• ⓑ log 3 ( 9 ) = 2 log 3 ( 9 ) = 2

First, identify the values of b , y , and x . b , y , and x . Then, write the equation in the form b y = x . b y = x .

Here, b = 6 , y = 1 2 , and   x = 6. b = 6 , y = 1 2 , and   x = 6. Therefore, the equation log 6 ( 6 ) = 1 2 log 6 ( 6 ) = 1 2 is equivalent to 6 1 2 = 6 . 6 1 2 = 6 .

Here, b = 3 , y = 2 , and   x = 9. b = 3 , y = 2 , and   x = 9. Therefore, the equation log 3 ( 9 ) = 2 log 3 ( 9 ) = 2 is equivalent to 3 2 = 9. 3 2 = 9.

• ⓐ log 10 ( 1, 000, 000 ) = 6 log 10 ( 1, 000, 000 ) = 6
• ⓑ log 5 ( 25 ) = 2 log 5 ( 25 ) = 2

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b , b , exponent x , x , and output y . y . Then we write x = log b ( y ) . x = log b ( y ) .

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

• 2 3 = 8 2 3 = 8
• 5 2 = 25 5 2 = 25
• 10 − 4 = 1 10,000 10 − 4 = 1 10,000

First, identify the values of b , y , and x . b , y , and x . Then, write the equation in the form x = log b ( y ) . x = log b ( y ) .

Here, b = 2 , b = 2 , x = 3 , x = 3 , and y = 8. y = 8. Therefore, the equation 2 3 = 8 2 3 = 8 is equivalent to log 2 ( 8 ) = 3. log 2 ( 8 ) = 3.

Here, b = 5 , b = 5 , x = 2 , x = 2 , and y = 25. y = 25. Therefore, the equation 5 2 = 25 5 2 = 25 is equivalent to log 5 ( 25 ) = 2. log 5 ( 25 ) = 2.

Here, b = 10 , b = 10 , x = − 4 , x = − 4 , and y = 1 10,000 . y = 1 10,000 . Therefore, the equation 10 − 4 = 1 10,000 10 − 4 = 1 10,000 is equivalent to log 10 ( 1 10,000 ) = − 4. log 10 ( 1 10,000 ) = − 4.

• ⓐ 3 2 = 9 3 2 = 9
• ⓑ 5 3 = 125 5 3 = 125
• ⓒ 2 − 1 = 1 2 2 − 1 = 1 2

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log 2 8. log 2 8. We ask, “To what exponent must 2 2 be raised in order to get 8?” Because we already know 2 3 = 8 , 2 3 = 8 , it follows that log 2 8 = 3. log 2 8 = 3.

Now consider solving log 7 49 log 7 49 and log 3 27 log 3 27 mentally.

• We ask, “To what exponent must 7 be raised in order to get 49?” We know 7 2 = 49. 7 2 = 49. Therefore, log 7 49 = 2 log 7 49 = 2
• We ask, “To what exponent must 3 be raised in order to get 27?” We know 3 3 = 27. 3 3 = 27. Therefore, log 3 27 = 3 log 3 27 = 3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log 2 3 4 9 log 2 3 4 9 mentally.

• We ask, “To what exponent must 2 3 2 3 be raised in order to get 4 9 ? 4 9 ? ” We know 2 2 = 4 2 2 = 4 and 3 2 = 9 , 3 2 = 9 , so ( 2 3 ) 2 = 4 9 . ( 2 3 ) 2 = 4 9 . Therefore, log 2 3 ( 4 9 ) = 2. log 2 3 ( 4 9 ) = 2.

Given a logarithm of the form y = log b ( x ) , y = log b ( x ) , evaluate it mentally.

• Rewrite the argument x x as a power of b : b : b y = x . b y = x .
• Use previous knowledge of powers of b b identify y y by asking, “To what exponent should b b be raised in order to get x ? x ? ”

Solving Logarithms Mentally

Solve y = log 4 ( 64 ) y = log 4 ( 64 ) without using a calculator.

First we rewrite the logarithm in exponential form: 4 y = 64. 4 y = 64. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

Solve y = log 121 ( 11 ) y = log 121 ( 11 ) without using a calculator.

Evaluating the Logarithm of a Reciprocal

Evaluate y = log 3 ( 1 27 ) y = log 3 ( 1 27 ) without using a calculator.

First we rewrite the logarithm in exponential form: 3 y = 1 27 . 3 y = 1 27 . Next, we ask, “To what exponent must 3 be raised in order to get 1 27 ? 1 27 ? ”

We know 3 3 = 27 , 3 3 = 27 , but what must we do to get the reciprocal, 1 27 ? 1 27 ? Recall from working with exponents that b − a = 1 b a . b − a = 1 b a . We use this information to write

Therefore, log 3 ( 1 27 ) = − 3. log 3 ( 1 27 ) = − 3.

Evaluate y = log 2 ( 1 32 ) y = log 2 ( 1 32 ) without using a calculator.

Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means log 2 ( x ) log 2 ( x ) . However, in mathematics it almost always means the common logarithm of 10. In other words, the expression log ( x ) log ( x ) often means log 10 ( x ) . log 10 ( x ) .

Definition of the Common Logarithm

A common logarithm is a logarithm with base 10. 10. We can also write log 10 ( x ) log 10 ( x ) simply as log ( x ) . log ( x ) . The common logarithm of a positive number x x satisfies the following definition.

For x > 0 , x > 0 ,

We read log ( x ) log ( x ) as, “the logarithm with base 10 10 of x x ” or “log base 10 of x . x . ”

The logarithm y y is the exponent to which 10 10 must be raised to get x . x .

Currently, we use log b ( x ) , lg ( x ) log b ( x ) , lg ( x ) as the common logarithm, lb ( x ) lb ( x ) as the binary logarithm, and ln ( x ) ln ( x ) as the natural logarithm. Writing lg ( x ) lg ( x ) without specifying a base is now considered bad form, despite being frequently found in older materials.

Given a common logarithm of the form y = log ( x ) , y = log ( x ) , evaluate it mentally.

• Rewrite the argument x x as a power of 10 : 10 : 10 y = x . 10 y = x .
• Use previous knowledge of powers of 10 10 to identify y y by asking, “To what exponent must 10 10 be raised in order to get x ? x ? ”

Finding the Value of a Common Logarithm Mentally

Evaluate y = log ( 1000 ) y = log ( 1000 ) without using a calculator.

First we rewrite the logarithm in exponential form: 10 y = 1000. 10 y = 1000. Next, we ask, “To what exponent must 10 10 be raised in order to get 1000?” We know

Therefore, log ( 1000 ) = 3. log ( 1000 ) = 3.

Evaluate y = log ( 1, 000, 000 ) . y = log ( 1, 000, 000 ) .

Given a common logarithm with the form y = log ( x ) , y = log ( x ) , evaluate it using a calculator.

• Press [LOG] .
• Enter the value given for x , x , followed by [ ) ] .
• Press [ENTER] .

Finding the Value of a Common Logarithm Using a Calculator

Evaluate y = log ( 321 ) y = log ( 321 ) to four decimal places using a calculator.

• Enter 321 , followed by [ ) ] .

Rounding to four decimal places, log ( 321 ) ≈ 2.5065. log ( 321 ) ≈ 2.5065.

Note that 10 2 = 100 10 2 = 100 and that 10 3 = 1000. 10 3 = 1000. Since 321 is between 100 and 1000, we know that log ( 321 ) log ( 321 ) must be between log ( 100 ) log ( 100 ) and log ( 1000 ) . log ( 1000 ) . This gives us the following:

Evaluate y = log ( 123 ) y = log ( 123 ) to four decimal places using a calculator.

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10 x = 500 10 x = 500 represents this situation, where x x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

We begin by rewriting the exponential equation in logarithmic form.

Next we evaluate the logarithm using a calculator:

• Enter 500 , 500 , followed by [ ) ] .
• To the nearest thousandth, log ( 500 ) ≈ 2.699. log ( 500 ) ≈ 2.699.

The difference in magnitudes was about 2.699. 2.699.

The amount of energy released from one earthquake was 8,500 8,500 times greater than the amount of energy released from another. The equation 10 x = 8500 10 x = 8500 represents this situation, where x x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Using Natural Logarithms

The most frequently used base for logarithms is e , e , the value of which is approximately 2.71828 2.71828 . Base e e logarithms are important in calculus and some scientific applications; they are called natural logarithms . The base e e logarithm, log e ( x ) , log e ( x ) , has its own notation, ln ( x ) . ln ( x ) .

Most values of ln ( x ) ln ( x ) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, ln 1 = 0. ln 1 = 0. For other natural logarithms, we can use the ln ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e e using the inverse property of logarithms.

Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e . e . We write log e ( x ) log e ( x ) simply as ln ( x ) . ln ( x ) . The natural logarithm of a positive number x x satisfies the following definition.

We read ln ( x ) ln ( x ) as, “the logarithm with base e e of x x ” or “the natural logarithm of x . x . ”

The logarithm y y is the exponent to which e e must be raised to get x . x .

Since the functions y = e x y = e x and y = ln ( x ) y = ln ( x ) are inverse functions, ln ( e x ) = x ln ( e x ) = x for all x x and e = ln ( x ) x e = ln ( x ) x for x > 0. x > 0.

Given a natural logarithm with the form y = ln ( x ) , y = ln ( x ) , evaluate it using a calculator.

• Press [LN] .

Evaluating a Natural Logarithm Using a Calculator

Evaluate y = ln ( 500 ) y = ln ( 500 ) to four decimal places using a calculator.

Rounding to four decimal places, ln ( 500 ) ≈ 6.2146 ln ( 500 ) ≈ 6.2146

Evaluate ln ( −500 ) . ln ( −500 ) .

Access this online resource for additional instruction and practice with logarithms.

• Introduction to Logarithms

6.3 Section Exercises

What is a base b b logarithm? Discuss the meaning by interpreting each part of the equivalent equations b y = x b y = x and log b x = y log b x = y for b > 0 , b ≠ 1. b > 0 , b ≠ 1.

How is the logarithmic function f ( x ) = log b x f ( x ) = log b x related to the exponential function g ( x ) = b x ? g ( x ) = b x ? What is the result of composing these two functions?

How can the logarithmic equation log b x = y log b x = y be solved for x x using the properties of exponents?

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base b , b , and how does the notation differ?

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base b , b , and how does the notation differ?

For the following exercises, rewrite each equation in exponential form.

log 4 ( q ) = m log 4 ( q ) = m

log a ( b ) = c log a ( b ) = c

log 16 ( y ) = x log 16 ( y ) = x

log x ( 64 ) = y log x ( 64 ) = y

log y ( x ) = −11 log y ( x ) = −11

log 15 ( a ) = b log 15 ( a ) = b

log y ( 137 ) = x log y ( 137 ) = x

log 13 ( 142 ) = a log 13 ( 142 ) = a

log ( v ) = t log ( v ) = t

ln ( w ) = n ln ( w ) = n

For the following exercises, rewrite each equation in logarithmic form.

4 x = y 4 x = y

c d = k c d = k

m − 7 = n m − 7 = n

19 x = y 19 x = y

x − 10 13 = y x − 10 13 = y

n 4 = 103 n 4 = 103

( 7 5 ) m = n ( 7 5 ) m = n

y x = 39 100 y x = 39 100

10 a = b 10 a = b

e k = h e k = h

For the following exercises, solve for x x by converting the logarithmic equation to exponential form.

log 3 ( x ) = 2 log 3 ( x ) = 2

log 2 ( x ) = − 3 log 2 ( x ) = − 3

log 5 ( x ) = 2 log 5 ( x ) = 2

log 3 ( x ) = 3 log 3 ( x ) = 3

log 2 ( x ) = 6 log 2 ( x ) = 6

log 9 ( x ) = 1 2 log 9 ( x ) = 1 2

log 18 ( x ) = 2 log 18 ( x ) = 2

log 6 ( x ) = − 3 log 6 ( x ) = − 3

log ( x ) = 3 log ( x ) = 3

ln ( x ) = 2 ln ( x ) = 2

For the following exercises, use the definition of common and natural logarithms to simplify.

log ( 100 8 ) log ( 100 8 )

10 log ( 32 ) 10 log ( 32 )

2 log ( .0001 ) 2 log ( .0001 )

e ln ( 1.06 ) e ln ( 1.06 )

ln ( e − 5.03 ) ln ( e − 5.03 )

e ln ( 10.125 ) + 4 e ln ( 10.125 ) + 4

For the following exercises, evaluate the base b b logarithmic expression without using a calculator.

log 3 ( 1 27 ) log 3 ( 1 27 )

log 6 ( 6 ) log 6 ( 6 )

log 2 ( 1 8 ) + 4 log 2 ( 1 8 ) + 4

6 log 8 ( 4 ) 6 log 8 ( 4 )

For the following exercises, evaluate the common logarithmic expression without using a calculator.

log ( 10 , 000 ) log ( 10 , 000 )

log ( 0.001 ) log ( 0.001 )

log ( 1 ) + 7 log ( 1 ) + 7

2 log ( 100 − 3 ) 2 log ( 100 − 3 )

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

ln ( e 1 3 ) ln ( e 1 3 )

ln ( 1 ) ln ( 1 )

ln ( e − 0.225 ) − 3 ln ( e − 0.225 ) − 3

25 ln ( e 2 5 ) 25 ln ( e 2 5 )

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

log ( 0.04 ) log ( 0.04 )

ln ( 15 ) ln ( 15 )

ln ( 4 5 ) ln ( 4 5 )

log ( 2 ) log ( 2 )

ln ( 2 ) ln ( 2 )

Is x = 0 x = 0 in the domain of the function f ( x ) = log ( x ) ? f ( x ) = log ( x ) ? If so, what is the value of the function when x = 0 ? x = 0 ? Verify the result.

Is f ( x ) = 0 f ( x ) = 0 in the range of the function f ( x ) = log ( x ) ? f ( x ) = log ( x ) ? If so, for what value of x ? x ? Verify the result.

Is there a number x x such that ln x = 2 ? ln x = 2 ? If so, what is that number? Verify the result.

Is the following true: log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? Verify the result.

Is the following true: ln ( e 1.725 ) ln ( 1 ) = 1.725 ? ln ( e 1.725 ) ln ( 1 ) = 1.725 ? Verify the result.

Real-World Applications

The exposure index E I E I for a camera is a measurement of the amount of light that hits the image receptor. It is determined by the equation E I = log 2 ( f 2 t ) , E I = log 2 ( f 2 t ) , where f f is the “f-stop” setting on the camera, and t t is the exposure time in seconds. Suppose the f-stop setting is 8 8 and the desired exposure time is 2 2 seconds. What will the resulting exposure index be?

Refer to the previous exercise. Suppose the light meter on a camera indicates an E I E I of − 2 , − 2 , and the desired exposure time is 16 seconds. What should the f-stop setting be?

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 − M 2 log I 1 I 2 = M 1 − M 2 where M M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. 8 How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

• 4 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.
• 5 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.
• 6 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.
• 7 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
• 8 http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.

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How To Solve Logarithmic Equations

Step By Step Video and practice Problems

more interesting facts

How To Solve Logarithmic Equations Video

What is the general strategy for solving log equations?

Answer: As the video above points out, there are two main types of logarithmic equations. Before you to decide how to solve an equation, you must determine whether the equation

• A) has a logarithm on one side and a number on the other
• B) whether it has logarithms on both sides

Example 1 Logarithm on one side and a number on the other

$$ log_4 x + log_4 8 = 3 $$

Step 1 Rewrite log side as single logarithm

$$ log_4 8x = 3 $$

Step 2 Rewrite as exponential equation

$$ 4^ 3 = 8x $$

Step 3 Solve the exponential equation

64 = 8x 8 = x

Example 2 Logarithm on both sides

Step 1 use the rules of logarithms to rewrite the left side and the right side of the equation to a single logarithm

Step 2 "cancel" the log

Step 3 solve the expression

Let's look at a specific ex $$ log_5 x + log_2 3 = log_5 6 $$

Step 1 rewrite both sides as single logs

$$ log_5 x + log_5 2 = log_5 6 \\ log_5 2x = log_5 6 $$

Step 2 "cancel" logs

$$ \color{Red}{ \cancel {log_5}} 2x = \color{Red}{ \cancel {log_5}} 6 \\ 2x = 6 $$

Step 2 Solve expression

Practice Problems

Solve the following equation: $ log_3 5 + log_3 x = log_3 15 $

Follow the steps for solving logarithmic equations with logs on both sides

rewrite both sides as single logs

$$ log_3 5x = log_3 15 $$

"cancel" logs

$$ \color{Red}{ \cancel{log_3}} 5x = \color{Red}{ \cancel{log_3}} 15 \\ 5x=15 $$

Solve expression

Solve the equation below: $ log_3 9 + log_3 x = 4 $

Follow the steps for solving logarithmic equations with a log on one side

Rewrite log side as single logarithm

$$ log_3 9x = 4$$

Rewrite as exponential equation

$$ 3^4 = 9x $$

Solve exponential equation

81 = 9 x 9 = x

Solve the following equation: $ 2log_3 5 + log_3 x = 3log_3 $

Follow the steps on how to solve equations with logs on both sides

rewrite both sides as single logs<

$ log_3 5^2 + log_3 x = log_3 5^3 \\ log_3 25 +log_3 x = log_3 125 \\ log_3 25x = log_3 125 $

$ \color{Red}{ \cancel{log_3}} 25x = \color{Red}{ \cancel{log_3}} 125 \\ 25x = 125 $

Solve the equation below: $ 2 log_2 4 + log_2 x = 5 $

$ 2 log_2 4 + log_2 x = 5 \\ log_2 4^2 = log_2 x = 5 \\ log_2 16 + log_2 x = 5 \\ log_2 16x = 5 $

32 =16x 2 = x

Solve the following equation: $ 2 log_3 5 + log_3 x = 3 log_3 5 $

$ log_3 5^2 + log_3 x + log_3 5^3 \\ log_3 25x + log_3 125 $

log 3 25x = log 3 5 3

$ \color{Red}{ \cancel{log_3}} 25x + \color{Red}{ \cancel{log_3}} 125 \\ 25x=125 $

$ \frac{25x}{25} = \frac{125}{25} \\ $

Solve the following equation: $2 log_3 7 - log_3 2x = log_3 98$

$ 2 log_3 7 - log_3 2x = log_3 98 \\ log_3 7^2 - log_3 2x = log_3 98 \\ log_3 49 - log_3 2x = log_3 98 \\ log_3 \frac{49}{2x} = log_3 98 $

$ \color{Red}{ \cancel{log_3}} \frac{49}{2x} = \color{Red}{ \cancel{log_3}} 98 \\ \frac{49}{2x} = 98 $

$ 49 = 196x \\ \frac{49}{196} = x \\ x = 49 $

Solve the following equation: $ 2 log_11 5 + log_11 x + log_11 2 = log_11 150 $

You know the deal. Just follow the steps for solving logarithmic equations with logs on both sides

rewrite as single logs

$ 2 log_11 5 + log_11 x + log_11 2 = log_11 150 \\ log_11 5^2 + log_11 2x = log_11 150 \\ log_11 25 + log_11 2x = log_11 150 \\ log_11 50x= log_11 150 $

2log 11 5 + log 11 x+ log 11 2 = log 11 150

$ \color{Red}{ \cancel{log_1}} 50x = \color{Red}{ \cancel{log_11}} 150 \\ 50x = 150 $

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Properties of logarithmic functions

Comparison of exponential function and logarithmic function, practice questions, solving logarithmic functions – explanation & examples.

Logarithms and exponents are two topics in mathematics that are closely related. Therefore it is useful we take a brief review of exponents.

An exponent is a form of writing the repeated multiplication of a number by itself. An exponential function is of the form f (x) = b y , where b > 0 < x and b ≠ 1. The quantity x is the number, b is the base, and y is the exponent or power.

For example , 32 = 2 × 2 × 2 × 2 × 2 = 2 2 .

On the other hand, the logarithmic function is defined as the inverse function of exponentiation. Consider again the exponential function f(x) = b y , where b > 0 < x and b ≠ 1. We can represent this function in logarithmic form as:

y = log b x

Then the logarithmic function is given by;

f(x) = log b x = y, where b is the base, y is the exponent, and x is the argument.

The function f (x) = log b x is read as “log base b of x.” Logarithms are useful in mathematics because they enable us to perform calculations with very large numbers.

How to Solve Logarithmic Functions?

To solve the logarithmic functions, it is important to use exponential functions in the given expression. The natural log or ln is the inverse of e . That means one can undo the other one i.e.

ln (e x ) = x

To solve an equation with logarithm(s), it is important to know their properties.

Properties of logarithmic functions are simply the rules for simplifying logarithms when the inputs are in the form of division, multiplication, or exponents of logarithmic values.

Some of the properties are listed below.

• Product rule

The product rule of logarithm states the logarithm of the product of two numbers having a common base is equal to the sum of individual logarithms.

⟹ log a  (p q) = log a  p + log a  q.

• Quotient rule

The quotient rule of logarithms states that the logarithm of the two numbers’ ratio with the same bases is equal to the difference of each logarithm.

⟹ log a  (p/q) = log a  p – log a q

The power rule of logarithm states that the logarithm of a number with a rational exponent is equal to the product of the exponent and its logarithm.

⟹ log a  (p q ) = q log a p

• Change of Base rule

⟹ log a p = log x p ⋅ log a x

⟹ log q p = log x p / log x q

• Zero Exponent Rule

Other properties of logarithmic functions include:

• The bases of an exponential function and its equivalent logarithmic function are equal.
• The logarithms of a positive number to the base of the same number are equal to 1.

log a  a = 1

• Logarithms of 1 to any base are 0.

log a  1 = 0

• Log a 0 is undefined
• Logarithms of negative numbers are undefined.
• The base of logarithms can never be negative or 1.
• A logarithmic function with base 10is called a common logarithm. Always assume a base of 10 when solving with logarithmic functions without a small subscript for the base.

Whenever you see logarithms in the equation, you always think of how to undo the logarithm to solve the equation. For that, you use an exponential function . Both of these functions are interchangeable.

The following table tells the way of writing and interchanging the exponential functions and logarithmic functions . The third column tells about how to read both the logarithmic functions.

Let’s use these properties to solve a couple of problems involving logarithmic functions.

Rewrite exponential function 7 2 = 49 to its equivalent logarithmic function.

Given 7 2 = 64.

Here, the base = 7, exponent = 2 and the argument = 49. Therefore, 7 2 = 64 in logarithmic function is;

⟹ log 7 49 = 2

Write the logarithmic equivalent of 5 3 = 125.

exponent = 3;

and argument = 125

5 3 = 125 ⟹ log 5 125 =3

Solve for x in log  3  x = 2

log  3  x = 2 3 2  = x ⟹ x = 9

If 2 log x = 4 log 3, then find the value of ‘x’.

2 log x = 4 log 3

Divide each side by 2.

log x = (4 log 3) / 2

log x = 2 log 3

log x = log 3 2

log x = log 9

Find the logarithm of 1024 to the base 2.

1024 = 2 10

log 2 1024 = 10

Find the value of x in log 2 ( x ) = 4

Rewrite the logarithmic function log 2 ( x ) = 4 to exponential form.

Solve for x in the following logarithmic function log 2 (x – 1) = 5.

Solution Rewrite the logarithm in exponential form as;

log 2 (x – 1) = 5 ⟹ x – 1 = 2 5

Now, solve for x in the algebraic equation. ⟹ x – 1 = 32 x = 33

Find the value of x in log x 900 = 2.

Write the logarithm in exponential form as;

Find the square root of both sides of the equation to get;

x = -30 and 30

But since, the base of logarithms can never be negative or 1, therefore, the correct answer is 30.

Solve for x given, log x = log 2 + log 5

Using the product rule Log b  (m n) = log b  m + log b  n we get;

⟹ log 2 + log 5 = log (2 * 5) = Log   (10).

Therefore, x = 10.

Solve log  x  (4x – 3) = 2

Rewrite the logarithm in exponential form to get;

x 2  = 4x – 3

Now, solve the quadratic equation. x 2  = 4x – 3 x 2  – 4x + 3 = 0 (x -1) (x – 3) = 0

Since the base of a logarithm can never be 1, then the only solution is 3.

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Logarithmic Equations Calculator

Get detailed solutions to your math problems with our logarithmic equations step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of logarithmic equations. This solution was automatically generated by our smart calculator:

Apply the formula: $a\log_{b}\left(x\right)$$=\log_{b}\left(x^a\right)$

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

 Intermediate steps

Rewrite the number $0$ as a logarithm of base $10$

Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$

For two logarithms of the same base to be equal, their arguments must be equal. In other words, if $\log(a)=\log(b)$ then $a$ must equal $b$

Multiply both sides of the equation by $x+6$

Any expression multiplied by $1$ is equal to itself

Move everything to the left hand side of the equation

Factor the trinomial $x^2-x-6$ finding two numbers that multiply to form $-6$ and added form $-1$

Break the equation in $2$ factors and set each equal to zero, to obtain

Solve the equation ($1$)

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $2$ from both sides of the equation

Canceling terms on both sides

Solve the equation ($2$)

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-3$ from both sides of the equation

Combining all solutions, the $2$ solutions of the equation are

Verify that the solutions obtained are valid in the initial equation

The valid solutions to the logarithmic equation are the ones that, when replaced in the original equation, don't result in any logarithm of negative numbers or zero, since in those cases the logarithm does not exist

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10.6: Solve Exponential and Logarithmic Equations

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Learning Objectives

By the end of this section, you will be able to:

• Solve logarithmic equations using the properties of logarithms
• Solve exponential equations using logarithms
• Use exponential models in applications

Be Prepared 10.13

Before you get started, take this readiness quiz.

Solve: x 2 = 16 . x 2 = 16 . If you missed this problem, review Example 6.46.

Be Prepared 10.14

Solve: x 2 − 5 x + 6 = 0 . x 2 − 5 x + 6 = 0 . If you missed this problem, review Example 6.45.

Be Prepared 10.15

Solve: x ( x + 6 ) = 2 x + 5 . x ( x + 6 ) = 2 x + 5 . If you missed this problem, review Example 6.47.

Solve Logarithmic Equations Using the Properties of Logarithms

In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.

If our equation has two logarithms we can use a property that says that if log a M = log a N log a M = log a N then it is true that M = N . M = N . This is the One-to-One Property of Logarithmic Equations .

One-to-One Property of Logarithmic Equations

For M > 0 , N > 0 , a > 0 , M > 0 , N > 0 , a > 0 , and a ≠ 1 a ≠ 1 is any real number:

If log a M = log a N , then M = N . If log a M = log a N , then M = N .

To use this property, we must be certain that both sides of the equation are written with the same base.

Remember that logarithms are defined only for positive real numbers. Check your results in the original equation. You may have obtained a result that gives a logarithm of zero or a negative number.

Example 10.38

Solve: 2 log 5 x = log 5 81 . 2 log 5 x = log 5 81 .

Try It 10.75

Solve: 2 log 3 x = log 3 36 2 log 3 x = log 3 36

Try It 10.76

Solve: 3 log x = log 64 3 log x = log 64

Another strategy to use to solve logarithmic equations is to condense sums or differences into a single logarithm.

Example 10.39

Solve: log 3 x + log 3 ( x − 8 ) = 2 . log 3 x + log 3 ( x − 8 ) = 2 .

Try It 10.77

Solve: log 2 x + log 2 ( x − 2 ) = 3 log 2 x + log 2 ( x − 2 ) = 3

Try It 10.78

Solve: log 2 x + log 2 ( x − 6 ) = 4 log 2 x + log 2 ( x − 6 ) = 4

When there are logarithms on both sides, we condense each side into a single logarithm. Remember to use the Power Property as needed.

Example 10.40

Solve: log 4 ( x + 6 ) − log 4 ( 2 x + 5 ) = − log 4 x . log 4 ( x + 6 ) − log 4 ( 2 x + 5 ) = − log 4 x .

Try It 10.79

Solve: log ( x + 2 ) − log ( 4 x + 3 ) = − log x . log ( x + 2 ) − log ( 4 x + 3 ) = − log x .

Try It 10.80

Solve: log ( x − 2 ) − log ( 4 x + 16 ) = log 1 x . log ( x − 2 ) − log ( 4 x + 16 ) = log 1 x .

Solve Exponential Equations Using Logarithms

In the section on exponential functions, we solved some equations by writing both sides of the equation with the same base. Next we wrote a new equation by setting the exponents equal.

It is not always possible or convenient to write the expressions with the same base. In that case we often take the common logarithm or natural logarithm of both sides once the exponential is isolated.

Example 10.41

Solve 5 x = 11 . 5 x = 11 . Find the exact answer and then approximate it to three decimal places.

Try It 10.81

Solve 7 x = 43 . 7 x = 43 . Find the exact answer and then approximate it to three decimal places.

Try It 10.82

Solve 8 x = 98 . 8 x = 98 . Find the exact answer and then approximate it to three decimal places.

When we take the logarithm of both sides we will get the same result whether we use the common or the natural logarithm (try using the natural log in the last example. Did you get the same result?) When the exponential has base e , we use the natural logarithm.

Example 10.42

Solve 3 e x + 2 = 24 . 3 e x + 2 = 24 . Find the exact answer and then approximate it to three decimal places.

Try It 10.83

Solve 2 e x − 2 = 18 . 2 e x − 2 = 18 . Find the exact answer and then approximate it to three decimal places.

Try It 10.84

Solve 5 e 2 x = 25 . 5 e 2 x = 25 . Find the exact answer and then approximate it to three decimal places.

Use Exponential Models in Applications

In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.

We will again use the Compound Interest Formulas and so we list them here for reference.

Compound Interest

For a principal, P , invested at an interest rate, r , for t years, the new balance, A is:

A = P ( 1 + r n ) n t when compounded n times a year. A = P e r t when compounded continuously. A = P ( 1 + r n ) n t when compounded n times a year. A = P e r t when compounded continuously.

Example 10.43

Jermael’s parents put $10,000 in investments for his college expenses on his first birthday. They hope the investments will be worth $50,000 when he turns 18. If the interest compounds continuously, approximately what rate of growth will they need to achieve their goal?

Try It 10.85

Hector invests $ 10,000 $ 10,000 at age 21. He hopes the investments will be worth $ 150,000 $ 150,000 when he turns 50. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal?

Try It 10.86

Rachel invests $ 15,000 $ 15,000 at age 25. She hopes the investments will be worth $ 90,000 $ 90,000 when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

We have seen that growth and decay are modeled by exponential functions. For growth and decay we use the formula A = A 0 e k t . A = A 0 e k t . Exponential growth has a positive rate of growth or growth constant, k k , and exponential decay has a negative rate of growth or decay constant, k .

Exponential Growth and Decay

For an original amount, A 0 , A 0 , that grows or decays at a rate, k , for a certain time, t , the final amount, A , is:

A = A 0 e k t A = A 0 e k t

We can now solve applications that give us enough information to determine the rate of growth. We can then use that rate of growth to predict other situations.

Example 10.44

Researchers recorded that a certain bacteria population grew from 100 to 300 in 3 hours. At this rate of growth, how many bacteria will there be 24 hours from the start of the experiment?

This problem requires two main steps. First we must find the unknown rate, k . Then we use that value of k to help us find the unknown number of bacteria.

Try It 10.87

Researchers recorded that a certain bacteria population grew from 100 to 500 in 6 hours. At this rate of growth, how many bacteria will there be 24 hours from the start of the experiment?

Try It 10.88

Researchers recorded that a certain bacteria population declined from 700,000 to 400,000 in 5 hours after the administration of medication. At this rate of decay, how many bacteria will there be 24 hours from the start of the experiment?

Radioactive substances decay or decompose according to the exponential decay formula. The amount of time it takes for the substance to decay to half of its original amount is called the half-life of the substance.

Similar to the previous example, we can use the given information to determine the constant of decay, and then use that constant to answer other questions.

Example 10.45

The half-life of radium-226 is 1,590 years. How much of a 100 mg sample will be left in 500 years?

This problem requires two main steps. First we must find the decay constant k . If we start with 100-mg, at the half-life there will be 50-mg remaining. We will use this information to find k . Then we use that value of k to help us find the amount of sample that will be left in 500 years.

Try It 10.89

The half-life of magnesium-27 is 9.45 minutes. How much of a 10-mg sample will be left in 6 minutes?

Try It 10.90

The half-life of radioactive iodine is 60 days. How much of a 50-mg sample will be left in 40 days?

Access these online resources for additional instruction and practice with solving exponential and logarithmic equations.

• Solving Logarithmic Equations
• Solving Logarithm Equations
• Finding the rate or time in a word problem on exponential growth or decay

Section 10.5 Exercises

Practice makes perfect.

In the following exercises, solve for x .

log 4 64 = 2 log 4 x log 4 64 = 2 log 4 x

log 49 = 2 log x log 49 = 2 log x

3 log 3 x = log 3 27 3 log 3 x = log 3 27

3 log 6 x = log 6 64 3 log 6 x = log 6 64

log 5 ( 4 x − 2 ) = log 5 10 log 5 ( 4 x − 2 ) = log 5 10

log 3 ( x 2 + 3 ) = log 3 4 x log 3 ( x 2 + 3 ) = log 3 4 x

log 3 x + log 3 x = 2 log 3 x + log 3 x = 2

log 4 x + log 4 x = 3 log 4 x + log 4 x = 3

log 2 x + log 2 ( x − 3 ) = 2 log 2 x + log 2 ( x − 3 ) = 2

log 3 x + log 3 ( x + 6 ) = 3 log 3 x + log 3 ( x + 6 ) = 3

log x + log ( x + 3 ) = 1 log x + log ( x + 3 ) = 1

log x + log ( x − 15 ) = 2 log x + log ( x − 15 ) = 2

log ( x + 4 ) − log ( 5 x + 12 ) = − log x log ( x + 4 ) − log ( 5 x + 12 ) = − log x

log ( x − 1 ) − log ( x + 3 ) = log 1 x log ( x − 1 ) − log ( x + 3 ) = log 1 x

log 5 ( x + 3 ) + log 5 ( x − 6 ) = log 5 10 log 5 ( x + 3 ) + log 5 ( x − 6 ) = log 5 10

log 5 ( x + 1 ) + log 5 ( x − 5 ) = log 5 7 log 5 ( x + 1 ) + log 5 ( x − 5 ) = log 5 7

log 3 ( 2 x − 1 ) = log 3 ( x + 3 ) + log 3 3 log 3 ( 2 x − 1 ) = log 3 ( x + 3 ) + log 3 3

log ( 5 x + 1 ) = log ( x + 3 ) + log 2 log ( 5 x + 1 ) = log ( x + 3 ) + log 2

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

3 x = 89 3 x = 89

2 x = 74 2 x = 74

5 x = 110 5 x = 110

4 x = 112 4 x = 112

e x = 16 e x = 16

e x = 8 e x = 8

( 1 2 ) x = 6 ( 1 2 ) x = 6

( 1 3 ) x = 8 ( 1 3 ) x = 8

4 e x + 1 = 16 4 e x + 1 = 16

3 e x + 2 = 9 3 e x + 2 = 9

6 e 2 x = 24 6 e 2 x = 24

2 e 3 x = 32 2 e 3 x = 32

1 4 e x = 3 1 4 e x = 3

1 3 e x = 2 1 3 e x = 2

e x + 1 + 2 = 16 e x + 1 + 2 = 16

e x − 1 + 4 = 12 e x − 1 + 4 = 12

In the following exercises, solve each equation.

3 3 x + 1 = 81 3 3 x + 1 = 81

6 4 x − 17 = 216 6 4 x − 17 = 216

e x 2 e 14 = e 5 x e x 2 e 14 = e 5 x

e x 2 e x = e 20 e x 2 e x = e 20

log a 64 = 2 log a 64 = 2

log a 81 = 4 log a 81 = 4

ln x = −8 ln x = −8

ln x = 9 ln x = 9

log 5 ( 3 x − 8 ) = 2 log 5 ( 3 x − 8 ) = 2

log 4 ( 7 x + 15 ) = 3 log 4 ( 7 x + 15 ) = 3

ln e 5 x = 30 ln e 5 x = 30

ln e 6 x = 18 ln e 6 x = 18

3 log x = log 125 3 log x = log 125

7 log 3 x = log 3 128 7 log 3 x = log 3 128

log 6 x + log 6 ( x − 5 ) = log 6 24 log 6 x + log 6 ( x − 5 ) = log 6 24

log 9 x + log 9 ( x − 4 ) = log 9 12 log 9 x + log 9 ( x − 4 ) = log 9 12

log 2 ( x + 2 ) − log 2 ( 2 x + 9 ) = − log 2 x log 2 ( x + 2 ) − log 2 ( 2 x + 9 ) = − log 2 x

log 6 ( x + 1 ) − log 6 ( 4 x + 10 ) = log 6 1 x log 6 ( x + 1 ) − log 6 ( 4 x + 10 ) = log 6 1 x

In the following exercises, solve for x , giving an exact answer as well as an approximation to three decimal places.

6 x = 91 6 x = 91

( 1 2 ) x = 10 ( 1 2 ) x = 10

7 e x − 3 = 35 7 e x − 3 = 35

8 e x + 5 = 56 8 e x + 5 = 56

In the following exercises, solve.

Sung Lee invests $ 5,000 $ 5,000 at age 18. He hopes the investments will be worth $ 10,000 $ 10,000 when he turns 25. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Alice invests $ 15,000 $ 15,000 at age 30 from the signing bonus of her new job. She hopes the investments will be worth $ 30,000 $ 30,000 when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Coralee invests $ 5,000 $ 5,000 in an account that compounds interest monthly and earns 7 % . 7 % . How long will it take for her money to double?

Simone invests $ 8,000 $ 8,000 in an account that compounds interest quarterly and earns 5 % . 5 % . How long will it take for his money to double?

Researchers recorded that a certain bacteria population declined from 100,000 to 100 in 24 hours. At this rate of decay, how many bacteria will there be in 16 hours?

Researchers recorded that a certain bacteria population declined from 800,000 to 500,000 in 6 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 24 hours?

A virus takes 6 days to double its original population ( A = 2 A 0 ) . ( A = 2 A 0 ) . How long will it take to triple its population?

A bacteria doubles its original population in 24 hours ( A = 2 A 0 ) . ( A = 2 A 0 ) . How big will its population be in 72 hours?

Carbon-14 is used for archeological carbon dating. Its half-life is 5,730 years. How much of a 100-gram sample of Carbon-14 will be left in 1000 years?

Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material form a 0.5 ml injection will be in the body in 24 hours?

Writing Exercises

Explain the method you would use to solve these equations: 3 x + 1 = 81 , 3 x + 1 = 81 , 3 x + 1 = 75 . 3 x + 1 = 75 . Does your method require logarithms for both equations? Why or why not?

What is the difference between the equation for exponential growth versus the equation for exponential decay?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

• Math Article

Logarithm Questions

Logarithm questions with answers are provided for students to solve them and understand the concept elaborately. These questions are based on the logarithm chapter of Class 9, 10 and 11 syllabi. Practising these problems will not only help students to score good marks in academic exams but also participate in competitive exams conducted at the state or national level, such as Maths Olympiad.

The logarithmic function is an inverse of the exponential function. It is defined as:

y=log a x, if and only if x=a y ; for x>0, a>0, and a≠1.

Natural logarithmic function: The log function with base e is called natural logarithmic function and is denoted by log e .

f(x) = log e x

The questions of logarithm could be solved based on the properties, given below:

Also, read:

• Logarithm Table

Questions on Logarithm with Solutions

1. Express 5 3 = 125 in logarithm form.

As we know,

a b = c ⇒ log a c=b

Log 5 125 = 3

2. Express log 10 1 = 0 in exponential form.

Given, log 10 1 = 0

By the rule, we know;

log a c=b ⇒ a b = c

3. Find the log of 32 to the base 4.

Solution: log 4 32 = x

(2 2 ) x = 2x2x2x2x2

log 4 32 =5/2

4. Find x if log 5 (x-7)=1.

Solution: Given,

log 5 (x-7)=1

Using logarithm rules, we can write;

5. If log a m=n, express a n-1 in terms of a and m.

6. Solve for x if log(x-1)+log(x+1)=log 2 1

Solution: log(x-1)+log(x+1)=log 2 1

log(x-1)+log(x+1)=0

log[(x-1)(x+1)]=0

Since, log 1 = 0

(x-1)(x+1) = 1

Since, log of negative number is not defined.

Therefore, x=√2

7. Express log(75/16)-2log(5/9)+log(32/243) in terms of log 2 and log 3.

Solution: log(75/16)-2log(5/9)+log(32/243)

Since, nlog a m=log a m n

⇒log(75/16)-log(5/9) 2 +log(32/243)

⇒log(75/16)-log(25/81)+log(32/243)

Since, log a m-log a n=log a (m/n)

⇒log[(75/16)÷(25/81)]+log(32/243)

⇒log[(75/16)×(81/25)]+log(32/243)

⇒log(243/16)+log(32/243)

Since, log a m+log a n=log a mn

⇒log(32/16)

8. Express 2logx+3logy=log a in logarithm free form.

Solution: 2logx+3logy=log a

Video Lesson

Logarithmic equations.

9. Prove that: 2log(15/18)-log(25/162)+log(4/9)=log2

Solution: 2log(15/18)-log(25/162)+log(4/9)=log2

Taking L.H.S.:

⇒2log(15/18)-log(25/162)+log(4/9)

⇒log(15/18) 2 -log(25/162)+log(4/9)

⇒log(225/324)-log(25/162)+log(4/9)

⇒log[(225/324)(4/9)]-log(25/162)

⇒log[(225/324)(4/9)]/(25/162)

⇒log(72/36)

⇒log2 (R.H.S)

10. Express log 10 (2) + 1 in the form of log 10 x.

Solution: log 10 (2)+1

11. Find the value of x, if log 10 (x-10)=1.

Solution: Given, log 10 (x-10)=1.

log 10 (x-10) = log 10 10

12. Find the value of x, if log(x+5)+log(x-5)=4log2+2log3

log(x+5)+log(x-5)=4log2+2log3

log(x 2 -25) = log2 4 +log3 2

log(x 2 -25) = log16+log9

log(x 2 -25)=log(16×9)

log(x 2 -25)=log144

x 2 -25=144

13. Solve for x, if (log 225/log15) = log x

Solution: log x = (log 225/log15)

log x = log 15 2 /log 15

log x = 2log 15/log 15

x=10×10

Practice Questions

• If log x = m+n and log y=m-n, express the value of log 10x/y 2 in terms of m and n.
• Express 3 -2 =1/9 in logarithmic form.
• Express log 10 0.01=-2 in exponential form.
• Find the logarithm of 1/81 to the base 27.
• Find x if log 7 (2x 2 -1)=2.

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How to Solve Natural Logarithms Problems? (+FREE Worksheet!)

In this blog post, you will learn more about Natural Logarithms and how to solve problems related to natural logarithms.

Related Topics

• How to Solve Logarithmic Equations
• How to Evaluate Logarithms
• Properties of Logarithms

Step by step guide to solve Natural Logarithms

• A natural logarithm is a logarithm that has a special base of the mathematical constant \(e\), which is an irrational number approximately equal to \(2.71\).
• The natural logarithm of \(x\) is generally written as ln \(x\), or \(\log_{e}{x}\).

Natural Logarithms – Example 1:

Solve the equation for \(x\): \(e^x=3\)

If \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))→ln(e^x)=ln(3) \)

Use log rule: \(\log_{a}{x^b}=b \log_{a}{x}\), then: \(ln(e^x)=x ln(e)→xln(e)=ln(3) \)

\(ln(e)=1\), then: \(x=ln(3) \)

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Natural logarithms – example 2:.

Solve equation for \(x\): \(ln(2x-1)=1\)

Use log rule: \(a=\log_{b}{b^a}\), then: \(1=ln⁡(e^1 )=ln⁡(e)→ln⁡(2x-1)=ln⁡(e)\)

When the logs have the same base: \(\log_{b}{f(x)}=\log_{b}{g(x)}\), then: \(f(x)=g(x)\)

then: \(ln(2x-1)=ln(e)\), then: \(2x-1=e→x=\frac{e+1}{2}\)

Natural Logarithms – Example 3:

Solve the equation for \(x\): \(e^x=5\)

If \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))→ln(e^x)=ln(5) \)

Use log rule: \(\log_{a}{x^b}=b \log_{a}{x}\), then: \(ln(e^x)=x ln(e)→xln(e)=ln(5) \)

\(ln(e)=1\), then: \(x=ln(5) \)

Natural Logarithms – Example 4:

Solve equation for \(x\): \(ln(5x-1)=1\)

Use log rule: \(a=\log_{b}{b^a}\), then: \(1=ln⁡(e^1 )=ln⁡(e)→ln⁡(5x-1)=ln⁡(e)\)

then: \(ln(5x-1)=ln(e)\), then: \(5x-1=e→x=\frac{e+1}{5}\)

Exercises to practice Natural Logarithms

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Solve each equation for \(x\)..

• \(\color{blue}{e^x=3}\)
• \(\color{blue}{e^x=4}\)
• \(\color{blue}{e^x=8}\)
• \(\color{blue}{ln x=6}\)
• \(\color{blue}{ln (ln x)=5}\)
• \(\color{blue}{e^x=9}\)
• \(\color{blue}{ln⁡(2x+5)=4}\)
• \(\color{blue}{ln(2x-1)=1}\)
• \(\color{blue}{x=ln 3}\)
• \(\color{blue}{x=ln 4,x=2ln⁡(2)}\)
• \(\color{blue}{x=ln 8,x=3ln⁡(2)}\)
• \(\color{blue}{x=e^6}\)
• \(\color{blue}{x=e^{e^5}}\)
• \(\color{blue}{x=ln 9,x=2ln⁡(3)}\)
• \(\color{blue}{x=\frac{e^4-5}{2}}\)
• \(\color{blue}{x=\frac{e+1}{2}}\)

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by: Effortless Math Team about 5 years ago (category: Articles , Free Math Worksheets )

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Logarithmic Word Problems

Log Probs Expo Growth Expo Decay

What are logarithm word problems?

Logarithmic word problems, in my experience, generally involve either evaluating a given logarithmic equation at a given point, or else solving an equation for a given variable; they're pretty straightforward.

Content Continues Below

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What real-world problems use logarithms?

The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses ( link ).

Note: While log-based word problems are, in my experience, pretty straightforward, their statements tend to be fairly lengthy. Expect to have to plow through an unusual amount of text before they get to the point.

• Chemists define the acidity or alkalinity of a substance according to the formula pH =  −log[H + ] where [H + ] is the hydrogen ion concentration, measured in moles per liter. Solutions with a pH value of less than 7 are acidic; solutions with a pH value of greater than 7 are basic; solutions with a pH of 7 (such as pure water) are neutral.

a) Suppose that you test apple juice and find that the hydrogen ion concentration is [H + ] = 0.0003 . Find the pH value and determine whether the juice is basic or acidic.

b) You test some ammonia and determine the hydrogen ion concentration to be [H + ] = 1.3 × 10 −9 . Find the pH value and determine whether the ammonia is basic or acidic.

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In each case, I need to evaluate the pH function at the given value of [H + ] . In other words, this exercise, despite all the verbiage, is just plug-n-chug.

Since no base is specified, I will assume that the base for this logarithm is 10 , so that this is the so-called "common" log. (I happen to know that 10 is indeed the correct base, but they should have specified.)

a) In the case of the apple juice, the hydrogen ion concentration is [H + ] = 0.0003 , so:

pH = −log[H + ]

= −log[0.0003]

= 3.52287874528...

This value is less than 7 , so the apple juice is acidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H + ] = 1.3 × 10 −9 , so:

= −log[1.3 × 10 −9 ] = 8.88605664769...

This value is more than 7 , so the ammonia is basic.

(a) The juice is acidic with a pH of about 3.5 , and (b) the ammonia is basic with a pH of about 8.9 .

When a logarithm is given without a base being specified, different people in different contexts will assume different bases; either 10 , 2 , or e . Ask now whether or not bases will be specified for all exercises, or if you're going to be expected to "just know" the bases for certain formulas, or if you're supposed to "just assume" that all logs without a specified base have a base of... [find out which one].

• "Loudness" is measured in decibels (abbreviated as dB). The formula for the loudness of a sound is given by dB = 10×log[I ÷ I 0 ] where I 0 is the intensity of "threshold sound", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound. For instance, a cat's purr is about 316 times as intense as threshold sound, for a decibel rating of:

dB = 10×log[I ÷ I 0 ]     = 10×log[ (316 I 0 ) ÷ I 0 ]     = 10×log[ 316 ]     = 24.9968708262...

...about 25 decibels.

Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a .22 rimfire rifle has an intensity of about I = (2.5 × 10 13 )I 0 , should you follow the rules and wear ear protection when practicing at the rifle range?

I need to evaluate the decibel equation at I = (2.5 × 10 13 )I 0 :

dB = 10log [ I ÷ I 0 ]     = 10log[ (2.5 ×10 13 )I 0 ÷ I 0 ]     = 10log[2.5 ×10 13 ]     = 133.979400087...

In other words, the squirrel gun creates a noise level of about 134 decibels. Since this is well above the level at which I can suffer hearing damage,

I should follow the rules and wear ear protection.

• Earthquake intensity is measured by the Richter scale. The formula for the Richter rating of a given quake is given by R = log[ I ÷ I 0  ] where I 0 is the "threshold quake", or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.

You have a seismograph set up at home, and see that there was an event while you were out that had an intensity of I = 989I 0 . Given that a heavy truck rumbling by can cause a microquake with a Richter rating of 3 or 3.5 , and "moderate" quakes have a Richter rating of 4 or more, what was likely the event that occurred while you were out?

To determine the probable event, I need to convert the intensity of the mystery quake into a Richter rating by evaluating the Richter function at I = 989I 0 :

R = log[ I ÷ I 0 ]     = log[ 989I 0 ÷ I 0 ]     = log[989]     = 2.9951962916...

A Richter rating of about 3 is not high enough to have been a moderate quake.

The event was probably just a big truck.

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CS50: Introduction to Computer Science

An introduction to the intellectual enterprises of computer science and the art of programming.

Associated Schools

Harvard School of Engineering and Applied Sciences

What you'll learn.

A broad and robust understanding of computer science and programming

How to think algorithmically and solve programming problems efficiently

Concepts like abstraction, algorithms, data structures, encapsulation, resource management, security, software engineering, and web development

Familiarity with a number of languages, including C, Python, SQL, and JavaScript plus CSS and HTML

How to engage with a vibrant community of like-minded learners from all levels of experience

How to develop and present a final programming project to your peers

Course description

This is CS50x , Harvard University's introduction to the intellectual enterprises of computer science and the art of programming for majors and non-majors alike, with or without prior programming experience. An entry-level course taught by David J. Malan, CS50x teaches students how to think algorithmically and solve problems efficiently. Topics include abstraction, algorithms, data structures, encapsulation, resource management, security, software engineering, and web development. Languages include C, Python, SQL, and JavaScript plus CSS and HTML. Problem sets inspired by real-world domains of biology, cryptography, finance, forensics, and gaming. The on-campus version of CS50x , CS50, is Harvard's largest course. 

Students who earn a satisfactory score on 9 problem sets (i.e., programming assignments) and a final project are eligible for a certificate. This is a self-paced course–you may take CS50x on your own schedule.

Instructors

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You'll Have No Words Seeing This Precious 2-Year-Old Genius Do Multiplication on AGT

Simon Cowell pleads, “Get us a calculator!” and asks Baby Dev to negotiate his next deal in this incredible AGT Season 19 Audition.

While most toddlers are stacking blocks and coloring with crayons, this pint-sized genius does addition, subtraction, and multiplication on national television.

How to Watch

Watch America’s Got Talent Tuesdays at 8/7c on NBC and next day on Peacock . 

Devan is the youngest Act ever to light up the America’s Got Talent stage, and in the Season 19 premiere, we can see why. The sweet boy knocked everyone’s socks off as he sailed through three different math equations that, to be honest, had the Judges stumped .

“We’re all sitting here thinking the same thing: get us a calculator,” said Simon Cowell after the performance, adding, “I now feel really stupid.”

RELATED:  Who Are the America's Got Talent Season 19 Judges?

Baby Dev’s father, Duane, a retired police officer, told NBC Insider he noticed something very special about his son early on: “At 4 months old, we realized his love for numbers.”

Duane explained that when he exposed his son to early learning shows, Baby Dev would suddenly cry when they switched away from math. “And when we turned it back, he would stop. We’re like, all right, he has this love and passion for numbers,” he said.

How old is Baby Devan?

Devan is 2 years old, the youngest-ever contestant on AGT . 

From the moment Baby Dev fist-bumped Terry Crews backstage before his Season 19 debut, AGT fans were charmed. The precious boy laughed, jumped, and squealed with delight after solving equations and politely saying “hello” and “thank you” to the Judges while on stage with his proud dad.

“His first word before mama and dada was seven,” Duane told the Judges. “That’s my lucky number,” exclaimed Simon. And it seemed to be Baby Dev’s, too. Three whiteboards were set up on stage — two for multiplication and one for addition — with Judges calling out random numbers. When Baby Dev quickly solved 7 x 9, there was a stunned silence before the crowd erupted in applause.

How did Devan develop his math skills?

Duane explained to NBC Insider how he nurtured his son’s penchant for math.

“At age 1, we had bought him a writing tablet. We would practice writing numbers with him from 1 to 10… and we would practice writing every day, and by 15 months old, he knew how to write numbers on his own.” After mastering addition, the math prodigy moved on to subtraction and his times tables.

“We can just be in bed, and he’ll be like… ‘Math, please. I wanna do math, please,’ and then he’ll go to his writing board and just do math equations,” said his dad, who noted that the little boy always loved counting everything. “Cars on the street, birds, trees… he has number magnets that he carries with him like it’s a toy.”

Baby Dev does not receive formal tutoring, and as one audience member pointed out, “He’s not even using his fingers” to count. However, he might have one strict requirement when calculating: Not being interrupted. As Baby Dev tackled a math problem and jotted down numbers, Howie Mandel loudly whispered, “We get to see the process.”

We can just be in bed, and he’ll be like… ‘Math, please. I wanna do math, please!’” Duane, Baby Dev's dad

The toddler shot him a look that could freeze water. After a long pause and some shushing directed toward Mandel, Baby Dev continued on and solved the final equation. He was unanimously voted on to the next round — but he may have something extra on his plate, too.

When Simon asked, “Devon, will you help me negotiate my next deal?” the little cutie responded, “Yes!”

Obviously this charming youngster moved on to the next round of the competition where fans are surely eager to see what mathematical wizardry he'll come up with next. 

Watch all-new episodes of  America’s Got Talent , airing Tuesdays at 8/7c on NBC   and streaming the next day on  Peacock .

-  Reporting by McKenzie Jean-Philippe

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How one agency uses AI to track and manage thousands of campaign assets — building its own library of 'collective knowledge'

• Saatchi & Saatchi struggled to find reusable assets in its vast backlog of marketing materials.
• The marketing agency started using an AI search engine called Lucy to simplify the process.
• This article is part of " CXO AI Playbook " — straight talk from business leaders on how they're testing and using AI.

For "CXO AI Playbook," Business Insider takes a look at mini case studies about AI adoption across industries, company sizes, and technology DNA. We've asked each of the featured companies to tell us about the problems they're trying to solve with AI, who's making these decisions internally, and their vision for using AI in the future.

Saatchi & Saatchi is a global communications and marketing agency that works with major brands , including Toyota and Tide. The agency, which is headquartered in London, is part of Publicis Communications , a hub of Publicis Groupe.

Situation analysis: What problem was the company trying to solve?

For each client and campaign, Saatchi & Saatchi creates a collection of assets, including images, videos, final presentations, strategic insights, and briefs. These items accumulate over the years as the agency continues working with clients, Jeremiah Knight, its chief operating officer, told Business Insider.

Often, teams need to review documents from past campaigns, such as when a company launches a product, he said, but the challenge is, "How do you find that stuff after you've created it?"

Sometimes, the people who worked on the original campaign can't remember where those documents are saved, or they might have left the agency, Knight added. Employees also don't always follow file-system hierarchies.

That can create a "needle in the haystack" situation to find necessary information, Knight said.

Key staff and partners

The agency realized that automation could help improve asset organization, Knight said, so he worked with the company's CEO and chief financial officer on a solution.

"Once you paint the picture of the problem and how valuable the solution might be, we all understood this would be a great thing for us to undertake," Knight added.

AI in action

About two years ago, Saatchi & Saatchi contracted with Lucy , an AI-powered search engine, to help find specific items in its system.

Lucy is integrated into the agency's Microsoft Teams chat function. Users can ask it questions through Teams, and Lucy searches the agency's files and sends results in Teams, Knight said.

To access the documents, users are directed to log in to the Lucy web interface. They can also search and find information directly through the web interface without going through Teams.

"It's sort of a library of everything of our collective knowledge," Knight said, including creative assets, campaign data, and other information.

To help train the artificial-intelligence model to produce the desired results, the agency hosted trainings on how to use Lucy and encouraged everyone to use it and provide feedback.

"The more you play with it, the more you use it, the better it gets," Knight said. "We had to get over that hurdle to make sure that they continuously use it and help us train the model."

Did it work, and how did leaders know?

Lucy has helped Saatchi & Saatchi index information, Knight said. So if someone forgets to use the correct filing structure or saves information in the wrong place, the tool can usually find it.

"Not only is that a time saver, it actually is helping direct people straight to the documents that are the most valuable to answer the question, 'How can we learn from what we've done before?'" he said.

The AI implementation is still a work in progress , Knight added: "This is one of the most exciting times to play and experiment and see what's possible, unlock so many things creatively and workflow-wise for different departments."

What's next?

Saatchi & Saatchi continues to leverage generative AI across its agency in multiple ways, such as creative conceptualizing , minimizing repetitive tasks, and analyzing data, Knight said.

He said the investment in the technology would continue as the agency sought to create efficiencies. In January, Publicis Groupe announced plans to invest $326 million in AI over the next three years, including a proprietary tool, CoreAI.

"AI helps with every single business problem," he said. "The less time you spend hunting for things, the faster you can create business value for your clients."

We want to hear from you. If you are interested in sharing your company's AI journey, email [email protected] .

• Main content

Solving the macOS Installer’s “Failed to Personalize” Error with New Firmware

Kudos to user LALicata on TidBITS Talk for sharing a macOS update solution that isn’t widely available on the Internet. Attempting to update an M2 16-inch MacBook Pro from macOS 13.6 Ventura to any subsequent version of Ventura or macOS 14.5 Sonoma always resulted in a “Failed to personalize” error.

This error pops up quickly in Internet searches, and several articles offer possible fixes, including many old standbys:

• Restart your Mac
• Make sure your Internet connection is working
• Clear sufficient space on your Mac
• Repair the boot drive with Disk Utility
• Employ the full installer from the Mac App Store
• Boot into Safe Mode before installing
• Install from macOS Recovery
• Use a bootable USB installer
• Erase the Mac from Recovery Assistant or iCloud before reinstalling

Those are all reasonable troubleshooting steps, but none solved the problem. Apple’s support reps kept focusing on the Internet connection under the assumption that the Mac couldn’t reach a necessary Apple server during the installation process. However, once LALicata mentioned that an M1 MacBook Air was able to download and install macOS updates with no problem, they focused on the Mac itself and eventually provided the solution, which was to restore the MacBook Pro to factory settings using Apple Configurator, which writes a fresh copy of the firmware to the Mac.

Firmware Updates

Typically, Mac firmware is updated whenever a new version of macOS is installed, but if something goes wrong in the process, the Mac can be left with outdated firmware. When automatic firmware updates fail, the solution is to “revive” or “restore” the Mac using another Mac running macOS 12 Monterey or later and a USB-C cable that supports data and charging, such as the Apple USB-C Charge Cable (Apple explicitly warns against trying to use a Thunderbolt 3 cable). Although Macs running Sonoma can update firmware using the Finder, Apple Configurator is necessary for Macs running Monterey or Ventura, and LALicata’s Apple rep said that this particular problem could be resolved only by restoring from Apple Configurator, not the Finder. (Reviving leaves your data in place and is worth trying first; restoring erases the Mac and reverts it to factory defaults.)

Before LALicata fixed the problem, when the MacBook Pro was still running macOS 13.6.4, its System Firmware Version was 10151.41.12. After using Apple Configurator and upgrading to macOS 14.5, the System Firmware Version jumped 10151.121.1, which is the latest version for all Apple silicon Macs . To check your Mac’s System Firmware Version, Option-click the Apple menu and choose System Information—you’ll see it in the Hardware Overview screen.

This scenario suggests another general avenue of Mac troubleshooting. If you’re having problems associated with startup or updating, compare your Mac’s current firmware version with the latest version. Howard Oakley’s excellent Silent Knight utility, which reports on the update status of various system settings, makes that easier.

The Answer Is Documentation

It’s easy to rail about the technical side of this problem. However, Apple certainly doesn’t want or expect this firmware corruption to happen and is undoubtedly working to prevent it from happening and work around it automatically if it does. But no one is perfect, and it’s impossible to predict every possible occurrence that could corrupt firmware during an update.

Instead, I’d argue that the problem here revolves around documentation. First, the error message is terrible. What does “Failed to personalize” mean (nothing, in at least this context, and not much in any I can imagine), and how is it related to firmware (it’s not)? I suspect that the firmware problem is sufficiently rare and unexpected that it’s triggering an unrelated message. Still, a better-worded error message would go a long way toward helping users (and support reps) figure out what’s going wrong.

The error condition might be rare, but it’s not unheard of, so the second problem is that Apple’s article about reviving and restoring Mac firmware doesn’t include the error message text as one of the symptoms of corrupted firmware. If it had, LALicata and other users could have found the solution with a quick search.

That’s partly why I’m writing this article—to seed Internet search engines with the “Failed to personalize” string and the solution. Anyone running into such a problem in the future should be able to find this article and jump to Apple’s instructions on reviving or restoring firmware. They may be somewhat involved and non-trivial to follow (it matters which USB-C port you use, for instance), but they’ll take far less time than working with remote support reps.

Ideally, no one will run into this problem again, but if someone does, I hope they find their way to the solution here.

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Comments About Solving the macOS Installer’s “Failed to Personalize” Error with New Firmware

Notable replies.

This is a welcome solution! I have had people write me about this over the years, and never had diagnosed what could have caused it—much less a solution!

Thanks so much, Adam. One of the many reasons I check the posts on this website!

I’ve an M1 Studio that has had the problem since shortly after I purchased it, now through three versions of the Mac OS. I am able to get around it by booting into recovery mode and then reinstalling the latest OS. I’ve tried many other solutions, including erasing and rebuilding, and twice using Configurator and a second Mac to perform a restore. Disappointingly, my interactions with Apple support 2 years ago were similar - they focused on issues interfering with the download of the update or network issues (despite the fact that I told them we have 2 other Macs in the household on the same network and neither has ever had this issue). Left it at the Genius Bar, they wiped the machine and installed Ventura, claimed they could then update to Sonoma without a problem. I reinstalled apps and data via a Time Machine backup and held my breath until the next Sonoma update appeared, and again got the “Failed to personalize” message, as I have for every update since. I guess this suggests the issue may be with some software I have on the computer, but Apple was unable to solve the problem in my case.

Thank you for this article on the “personalize” problem. I had to deal with it on my iMac. I couldn’t find guidance using that term, but I eventually resolved it through the Recovery route. Jim Wheelis

Running Monterrey (12.7.5) on an 2019 Retina 5K iMac and Ventura on M1 MBPro Ultra and M2 Air.

App Store says MacOS 14 required for Configurator. Haven’t tried downloading with the laptops running Ventura yet.

Any way to get around this?

Maybe one solution is to install MacOS Sonoma on an USB stick or external drive. Then boot from that drive to get Configurator and try to update the firmware. Another solution may be (have not tested it) is to use Configurator from your Macbook Air conected to the iMac with an Apple USB charge cable.

Many thanks for this article! I have been struggling with this issue for months on a MacMini 2018, i.e. still with an Intel processor. The only way to update system software (now Sonoma) was in recovery mode. I am looking forward to use this solution.

This was an interesting article; thank you. I checked, and my MacBook Air M1 (2020) running macOS Big Sur 11.7.10 has firmware version 10151.1.1 rather than 10151.121.1. Since I have lost most of the technical expertise that I had years ago, I would like to ignore the discrepancy and do nothing. FWIW, I installed 11.7.10 last September and have experienced no (new) problems since then.

There are three recurring issues with the MBA, and I wonder if the firmware could be causing or contributing to any of them.

First, Time Machine will not complete a backup, either to a Time Capsule or an external (spinning) disk. It used to, but stopped working two or three macOS updates ago. It does seem to back up everything from ~, and I suspect that the issue is permissions on some file that is near the end of a backup session.

Second, keystrokes are sometimes doubled. In other words, I’ll type a character and it will appear twice. Again, this has been going on for a long time. It is rare but annoyingly frequent, and it happens on both the external and internal keyboard.

Third, the external display will sometimes go blank for about two to six seconds. This has happened 22 times so far this calendar year (and like the other issues, has been occurring for years), sometimes twice in one day and one time it didn’t happen for 35 days. I cannot detect any common trigger.

So, what is the downside to doing nothing (and staying at firmware 10151.1.1, at least until I get around to upgrading macOS to something more modern)? Is it likely that any of the three issues is caused or aggravated by the out-of-date firmware? Thanks for any guidance.

And @lalicata deserves the lion’s share of credit for tracking it down and reporting the experience here!

There’s no way to tell, but a combination of a very old version of macOS and old firmware could explain it. Unless you have an important reason to stick with Big Sur, like an app that won’t run on a later version, I can’t see any reason not to upgrade to Sonoma.

Jon Lindemann

3 June 2024: Jon, I did NOT want to go to 14.5 but the Apple Engineers were quite sure that there was absolutely no way to fix my problem unless I went to 14.5. One did let it slip that in the future, do not discard the Configurator app that works with that system software. It was never said but I got the feeling that each major software jump gets its own configurator. Lee

Hi. This is Lee. I tried this with an Apple Engineer on Monday when I did not have the right cable on hand, Did not work. No amount of petting, stroking, talking nice, yelling, screaming, cursing, pleading, weeping, or begging could get the Configurator2 app to recognize the system software on the USB stick AND make active the “restore” command. I think, but am not sure at all that the firmware that is being updated on the faulty Mac must be read from a well behaving Mac. I am SWAGGING this.

Well, I don’t know. I’m using Excel version 16.16.27, and it’s a bit flaky (editing conditional formatting almost always causes a crash), but it’s a lot more important to me than trying to fix any of the three annoyances I mentioned, especially if the fix only “might” take care of the problem. Does anyone know if my 2018 (copyright date from About Excel) version of Excel would work with Sonoma? Ventura?

My plan is to get a new MBA and install an up-to-date version of Office and try to make sure that everything works, then upgrade macOS on the old MBA and install an up-to-date version of Office, and then retire my MacBook Air from early 2015.

Granted, it’s not getting any updates anymore, but I couldn’t care less. In fact, I prefer the peace and quiet that comes with not getting constant MS updater nags. I’m never going to allow Excel to dao anythign that could endanger my system anyway.

I had this with my M1 and a Dell 4K display I was using external retina display. I believe the 60Hz 4K USB-C to HDMI dongle I was using wasn’t quite up to the task, as I’d read other reports of the issue on the Amazon listing for that adapter. It felt to me like a bad clock or refresh sync, but I too can’t explain the sporadic nature of it happening.

Anyway, I stopped using that display as it wasn’t really mine. My wife uses it with her late-2013 Intel MBP and has no issues. These days I use a non-retina display, for my sins.

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Participants.

The 7 Best AI Tools to Help Solve Math Problems

How do you make seven even? Use these tools to solve the big math problems in life.

Quick Links

The test questions, wolframalpha, microsoft mathsolver.

While OpenAI's ChatGPT is one of the most widely known AI tools, there are numerous other platforms that students can use to improve their math skills.

I tested seven AI tools on two common math problems so you know what to expect from each platform and how to use each of them.

I used two math problems to test each tool and standardize the inputs.

• Solve for b: (2 / (b - 3)) - (6 / (2b + 1)) = 4
• Simplify the expression: (4 / 12) + (9 / 8) x (15 / 3) - (26 / 10)

These two problems give each AI tool a chance to show reasoning, problem-solving, accuracy, and how it can guide a learner through the process.

Thetawise provides more than simple answers; you can also opt to have the AI model tutor you by sharing a detailed step-by-step breakdown of the solution. Using the platform is fairly straightforward, given that all you need to do is navigate to the platform and key in the math problem at hand. Alternatively, you can even upload a photo of the math problem onto the platform, and the AI will analyze the image and provide you with an answer.

The AI platform gave us a step-by-step breakdown of the problem:

It resulted in the answer:

While the answer is correct, the tool also provides further options for students to generate a more detailed breakdown of the steps or ask more specific questions.

WolframAlpha is an AI tool capable of solving advanced arithmetic, calculus, and algebra equations. While WolframAlpha's free version provides you with a direct answer, the paid version of the tool generates step-by-step solutions. If you want to make the best use of WolframAlpha's capabilities, you can sign up for the Pro version, which costs $5 per month for the annual plan if you're a student.

As expected, Wolfram Alpha solved both problems, showcasing its ability to handle different problems and provide precise answers quickly.

Julius works pretty similarly to the other AI tools on this list. That said, the highlight of this platform is that it has a built-in community forum, which users can use to discuss their prompts, results, or even issues they might be facing with the platform. Its active user base helps you quickly exchange ideas and receive feedback or advice. The platform's default version uses a combination of GPT-4 and Calude-3, based on whichever model best suits the prompt you input.

We tested the platform's accuracy by submitting the same problems that we did with the other AI tools. When submitting your prompt, you have the option of typing your question or uploading an image or a Google Sheet.

Julius provided correct solutions and offered options to help users verify the solution.

One of the oldest AI platforms, Microsoft's MathSolver is a great option if you want a tool capable of providing free step-by-step solutions to calculus, algebra, and other math problems. Here's how it fared when we submitted our math problems.

Microsoft's MathSolver provided the correct answers, and you can view the steps to the solution, take a quiz, solve similar problems, and more. This can be a great way to practice and perfect your understanding of different concepts.

Symbolab allows you to practice your math skills via quizzes, track your progress, and provide solutions to mathematical problems of different types, including calculus, fractions, trigonometry, and more. You can also use the Digital Notebook feature to keep track of any math problems you solve and share them with your friends. Another highlight of this platform is that educators can use the tool to create a virtual classroom, generate assessments, and share feedback, among other things.

The platform not only displays the answer but also lets you view a breakdown of the steps involved in solving the problem. You can also share the answers and steps via email or social media or print them for reference.

Anthropic launched its Claude 3 AI models in March 2024. Anthropic stated that Claude Opus, the most advanced Claude 3 model, outperforms comparable AI tools on most benchmarks for AI systems, including basic mathematics, undergraduate-level expert knowledge, and graduate-level expert reasoning. To test the platform's accuracy and ease of use, we submitted our two math problems. Here's how the platform performed:

While Claude initially got the answer wrong, probing it and requesting further clarification led to a correct solution.

Remember that we used the free version of Claude to solve this problem; subscribing to Opus (its more advanced model) is recommended if you want to take advantage of Claude's more advanced problem-solving capabilities.

Given that Claude got the previous problem wrong, our second, more basic fraction-based problem will indicate if the AI's performance was an anomaly or part of a consistent pattern.

As you can see, Claude correctly solved this problem and provided a detailed step-by-step breakdown of how it arrived at the answer.

GPT-4 can solve problems with far greater accuracy than its predecessor, GPT-3.5. If you're using the free version of ChatGPT, you'll likely only have access to GPT 3.5 and GPT-4o . However, for $20 per month, you can subscribe to the Plus model, which gives you access to GPT-4 and allows you to input five times the number of messages per day compared to the free version. That said, let's check how it performs with math problems.

In both cases, GPT-4o provided the correct answer with a detailed breakdown of the steps. While the platform is free, unlike other models, it does not have a quiz feature or a community forum.

These AI tools offer unique features and capabilities that make them a good option for math problems. Ultimately, the best way to pick a tool is by testing different models to determine which platform best fits your preferences and learning needs.

Android Police

7 most common smart home problems and how to solve them.

Glitches with your smart home products can result in inconvenience and frustration

We live in an era of digital transformation, and the concept of a "smart home" is becoming popular due to its convenience and efficiency. From security to lights and temperature, these technologically advanced homes are equipped with interconnected smart devices to make our lives comfortable and easier. Like other electronics, they can have issues.

Whether you have problems with a smart bulb, fan, security camera, robot vacuum, thermostat, or smart display from Amazon or Google, this guide goes over common smart home problems and practical solutions to overcome them. Then, you can navigate your smart home ecosystem without errors.

How to set up Alexa smart home automation

1 compatibility issues.

There's no shortage of smart home manufacturers and vendors. While Amazon, Google, Apple, Philips, and other leading companies have agreed upon the Matter protocol , hundreds of devices still use Z-wave, Zigbee, Wi-Fi, and other standards.

If your preferred devices use different communication protocols, they may not work with each other properly. When you add products to your shopping list, check their compatibility with your existing devices and platforms to avoid such scenarios. Most manufacturers mention compatibility on the packaging.

2 Broken connectivity

Smart home products rely on an active internet connection to function properly. If all your smart home products act up, your home's Wi-Fi connection is the main culprit. Restart the router and try again.

Weak internet connectivity can also result in inconveniences. Place the router near your devices for a stable connection. Physical barriers (like walls and large furniture) and Bluetooth devices (such as microwaves and ovens) also break Wi-Fi connectivity. Think twice before setting up a router at home so that it can communicate with all smart home devices properly.

3 Battery drain

Several smart home products rely on batteries to function correctly. When these batteries run low, the device doesn't function properly. Similarly, some devices, such as security cameras, smart displays, and more, must be connected to a power source all the time.

Connecting too many devices to the same power source may overload the source and cause insufficient voltage and power supply. Check the battery life on these devices occasionally and make sure they aren't placed in weak Wi-Fi zones. A spotty connection may force it to work harder, resulting in battery drain.

4 Automation and Routine failures

Most smart home devices are compatible with voice assistants like Alexa, Siri, and Google Assistant . You can use these assistants to set up automation and routines via a single voice command. However, the entire command execution isn't always smooth. Multiple factors can break your automation rules and routines.

Network outages, incorrect settings, a spotty internet connection, or a sync delay can break your set routines. You can fix the network connection, delete and create automation again, update your devices to the latest version, and wait for a specific company to fix the network outage. Also, avoid setting up too many devices in a single automation. It should bring down the routine failure rate.

5 Issues with the voice assistant

Support for voice assistants plays a crucial role in a smooth, smart home setup. Many owners have complained about their preferred voice assistants not understanding their commands accurately. It could be due to background noise, different accents, speech patterns, and context variations that can result in misinterpreted commands.

Give commands with clarity. For example, instead of saying, "Turn on the fans," say, "Turn on the fans in the drawing room." Avoid using slang, technical terms, and unusual vocabulary in your voice commands.

6 Hardware failures

Hardware failures are common in smart home products. Sometimes, you may run into a damaged circuit board, blown fuses, capacitors, or dead batteries on a smart home product. There isn't much you can do when dealing with physical defects.

If possible, replace the malfunctioning part. You can also hire a professional to take a closer look. If your device is broken beyond repair, purchase a new unit to continue your seamless smart home setup.

7 Security concerns

Smart home devices like security cameras, smart displays, and more constantly monitor movements. These companies also collect large amounts of data to improve their products. Unauthorized access to your smart devices can result in a nightmare. Avoid setting up weak passwords on your home Wi-Fi network.

Also, install the latest firmware updates to enjoy security patches. Avoid third-party apps and use official apps to manage your smart devices. When a manufacturer updates its privacy policy, read it before agreeing to it.

How to make your own theft-deterrent system with smart home products

Get your smart home up and running in seconds.

Constant errors with smart home products can spoil your ideal setup in no time. Before calling customer care, troubleshoot the problem using the suggestions above. If you live in the Android ecosystem and plan to invest in more smart home products, glance over our dedicated buying guide to learn the top recommendations.

Three innovations to solve hotel staffing shortages

During the COVID-19 closures, hotel staff around the world were faced with the precarious prospect of staying home. For instance, in the United States, 70 percent of hotel staff were furloughed or laid off. 1 “AHLA data shows 70 percent of hotel employees laid off or furloughed,” TSNN, May 4, 2020. Many hospitality workers reevaluated their career paths and work-life balance as they braved the pandemic—since then, these labor pools have shed their concierge badges and room service trolleys for jobs that offer more flexibility or higher pay, such as retail, e-commerce, or warehousing.

Hotels are now battling to restore their staff complement to pre-pandemic levels. A recent survey of 200 hotels conducted by the American Hotel & Lodging Association (AHLA) revealed that 87 percent of hotels in the United States do not have enough staff, and 36 percent of the respondents cited severe staff shortages. For 43 percent of hotels, housekeeping roles were singled out as being the most seriously understaffed. AHLA points out that similar staff shortages extend to hotels across the United States as employment in the hotel industry decreased by almost 400,000 jobs between February 2020 and August 2022, with more than 115,000 job openings yet unfilled. 2 “87% of surveyed hotels report staffing shortages,” American Hotel & Lodging Association, October 3, 2022.

Nevertheless, travelers are returning in full force , leaving hotels with no choice but to operate at pre-pandemic occupancy with fewer staff members. The staff shortage may provide an opportunity for hotels to reframe the problem, and think creatively about roles and staffing.

This article highlights the value of adopting a flexible and agile staffing model that aligns with dynamic demand. This approach is marked by three innovations that could improve operational effectiveness and employee job satisfaction, without compromising service quality.

While the article presents success stories and positive outcomes as a result of implementing these innovations, it comes with a caveat that staffing remains a complex issue and the human element should not be glossed over. In a time of staff shortages, employee satisfaction (and therefore retention) is more important than ever.

Would you like to learn more about our Travel, Logistics & Infrastructure Practice ?

Innovation 1: using standardized, activity-based metrics to forecast frontline coverage needs.

Many hotels and resorts base their staffing on average weekly occupancy. This does not, however, reflect real need, as occupancy often fluctuates from day to day and week to week. And averages, by nature, flatten out peaks and troughs.

Take the example of a golf resort: When comparing average occupancy to actual occupancy, it is clear that weekly average numbers do not adequately reflect peak days or account for fluctuations day to day or week to week (Exhibit 1).

One solution could be to expand overall occupancy metrics to include specific measurements, like number of check-ins and check-outs, or number of tables seated at the hotel restaurant. Leveraging these metrics to match staffing to daily need (or even hourly need) enables hotels to define a standard set of productivity metrics for each type of role, which can be applied consistently as demand fluctuates.

A hotel could, for example, optimize staffing by assigning one room attendant for every 15 check-outs. Cleaning and laundry could also be scheduled to serve only the forecast number of check-outs per day. Where management ownership is shared across properties, further coverage can be achieved by standardizing activity-based metrics across all properties.

One hotel shifted from using weekly averages to staffing according to peaks in occupancy. By scheduling an additional front-desk employee for hours with high check-out volumes, this innovation reduced the total labor hours needed by about 10 percent. This also created better work-life balance for front-desk staff. One front-desk employee had previously left the industry to meet her childcare needs. She later returned to her position at the hotel because she could now work the new part-time shift during peak check-out periods. This freed up time for her to complete her work day, and still be home by the time her children returned from school.

Innovation 2: Redesigning roles to combine jobs

Traditional hotel staffing involves distinct roles with clear divisions between managers and frontline employees. And each discipline—such as housekeeping, front-desk, and maintenance—has separate promotion paths.

To build resilience in times of staff shortages, hotels could redesign roles so that fewer people are needed to perform the same number of tasks. This could involve combining similar roles, or cross-training staff so that they can switch roles.

For instance, when combining roles, housekeeping management could be combined with front-desk management to form a single set of duties. Hotels could also introduce a player-coach model where one role involves supervising and performing work, such as a housekeeping supervisor who manages a team and cleans rooms.

Staff can also cover multiple roles as needed. Night-shift roles could be transformed by combining low-touch housekeeping duties with front-desk coverage during quieter times.

Redesigned roles could potentially improve employee satisfaction by empowering staff to explore new career paths within the hotel’s operations. Combined roles build skills across disciplines—for example, supporting a housekeeper to train and become proficient in some maintenance areas, or a front-desk associate to build managerial skills.

How to ‘ACE’ hospitality recruitment

Innovation 3: creating job networks across properties.

Where management or ownership is shared across properties, roles may be staffed to cover a network of sites, rather than individual hotels—especially where a hotel group may already have centralized administration for functions such as legal, accounting, and human resources.

This approach could be effective for various roles including office support, frontline staff, specialized positions, and management. In specialized roles such as maintenance, security, or events planning, a single person covering multiple properties can create value as needed. Activities staff, for instance for kids’ clubs, could also be shared. Frontline roles could be networked across properties, without sacrificing customer service. For example, a concierge or front-desk employee could alternate shifts between two nearby locations, or two different types of properties with different needs. A resort may need check-out staff later in the day whereas a hotel that caters to business travelers would need more staff in the mornings. Managers could be pooled to lead two smaller teams at two properties, instead of being focused on one.

Combining innovations to elevate operations

Though each intervention could alleviate some of the post-pandemic staff shortage issues, a combination of all three innovations could bring about a considerable reduction in weekly staff hours for a hotel property.

To illustrate, at one resort, weekly hours could be reduced by up to 18 percent by applying these three staffing innovations, specifically to front-desk and housekeeping roles. In this instance, using hourly demand to plan for staff coverage could have the greatest impact (Exhibit 2).

In many cases, employee engagement and retention have increased at hotels where all three measures have been adopted. For example, a property group under shared ownership adopted all three measures during the past two years which led to new ways for employees to grow and develop. A housekeeper who never imagined himself in a managerial role is now a shift supervisor, and employees who learned English while in back-of-house roles have embarked on training to pick up new front-of-house shifts.

These successes bode well for understaffed hotels, as McKinsey’s 2020 Employee Experience survey shows that employees who report having a positive experience at work are 16 times more engaged, and are eight times more likely to remain with their employer, than employees who report a negative experience.

Accordingly, hotel owners could make efforts to engage employees when it comes to staffing innovation by exploring, testing, and experimenting to find what works best for their employees as well as for their business needs and operating environment.

While these three innovations, alone or in combination, can lead to operational improvements, they may be easier to implement at some types of hotels than others. For instance, resorts may find it easier to provide employees with new career paths, cross-train staff, or combine similar roles. Hotel-chain operators may find it easier to share roles across locations. Many smaller, limited-service hotels have already implemented job sharing, so this particular innovation may not offer much value. Hotel owners may need to assess what would likely work best for them and their particular business model and context.

As hotel occupancy continues to recover post-pandemic, hotels have the opportunity to think innovatively about staffing. By adopting a more flexible staffing model based on dynamic demand, hotels can deliver pre-pandemic levels of service even if they are short staffed. Of course, the key to implementing any staffing innovation successfully is to keep an eye on employee engagement, and ensure that any change is good for people, and the business.

Ryan Mann is a partner in McKinsey’s Chicago office, Esteban Ramirez is a manager of client capabilities in the Dallas office, and Matthew Straus is an associate partner in the Carolinas office.

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