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Inverse Properties of Logarithms

Simplify expressions using two properties of inverse logs

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Inverse Properties of Logarithmic Functions

If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation $y' = y$ is the general function $y = Ce^x$ . What is the inverse of this function?

Guidance

By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed (see the Inverse Functions concept), they equal $x$ . Therefore, if $f(x)=b^x$ and $g(x)=\log_b x$ , then:

$f \circ g=b^{\log_b x}=x$ and $g \circ f =\log_b b^x=x$

These are called the Inverse Properties of Logarithms.

Example A

Find:

a) $10^{\log 56}$

b) $e^{\ln6} \cdot e^{\ln2}$

Solution: For each of these examples, we will use the Inverse Properties.

a) Using the first property, we see that the bases cancel each other out. $10^{\log 56}=56$

b) Here, $e$ and the natural log cancel out and we are left with $6 \cdot 2=12$ .

Example B

Find $\log_4 16^x$

Solution: We will use the second property here. Also, rewrite 16 as $4^2$ .

$\log_4 16^x=\log_4 (4^2)^x=\log_4 4^{2x}=2x$

Example C

Find the inverse of $f(x)=2e^{x-1}$ .

Solution: See the Finding the Inverse concept for the steps on how to find the inverse.

Change $f(x)$ to $y$ . Then, switch $x$ and $y$ .

$& y=2e^{x-1} \\& x=2e^{y-1}$

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.

$& \frac{x}{2}=e^{y-1} \\& \ln \left(\frac{x}{2}\right)= \ln e^{y-1}$

Recall the Inverse Properties from earlier in this concept. $\log_b b^x=x$ ; applying this to the right side of our equation, we have $\ln e^{y-1}=y-1$ . Solve for $y$ .

$& \ln \left(\frac{x}{2}\right)=y-1 \\& \ln \left(\frac{x}{2}\right)+1=y$

Therefore, $\ln \left(\frac{x}{2}\right)+1$ is the inverse of $2e^{y-1}$ .

Intro Problem Revisit Switch x and y in the function $y = Ce^x$ and then solve for y .

$x = Ce^y\\\frac{x}{C} = e^y\\ln \frac{x}{C} = ln (e^y)\\ln \frac{x}{C} = y$

Therefore, the inverse of $y = Ce^x$ is $y = ln \frac{x}{C}$ .

Guided Practice

1. Simplify $5^{\log_5 6x}$ .

2. Simplify $\log_9 81^{x+2}$ .

3. Find the inverse of $f(x)=4^{x+2}-5$ .

1. Using the first inverse property, the log and the base cancel out, leaving $6x$ as the answer.

$5^{\log_5 6x}=6x$

2. Using the second inverse property and changing 81 into $9^2$ we have:

$\log_9 81^{x+2} &= \log_9 9^{2(x+2)} \\&= 2(x+2) \\&= 2x+4$

3. Follow the steps from Example C to find the inverse.

$f(x) &= 4^{x+2}-5 \\y &= 4^{x+2}-5 \\x &= 4^{y+2}-5 \\x+5 &= 4^{y+2} \\\log_4 (x+5) &= y+2 \\\log_4 (x+5)-2 &= y$

Vocabulary

Inverse Properties of Logarithms
$\log_b b^x=x$ and $b^{\log_b x}=x, b \ne 1$

Practice

Use the Inverse Properties of Logarithms to simplify the following expressions.

1. $\log_3 27^x$
2. $\log_5 \left(\frac{1}{5}\right)^x$
3. $\log_2 \left(\frac{1}{32}\right)^x$
4. $10^{\log(x+3)}$
5. $\log_6 36^{(x-1)}$
6. $9^{\log_9(3x)}$
7. $e^{\ln(x-7)}$
8. $\log \left(\frac{1}{100}\right)^{3x}$
9. $\ln e^{(5x-3)}$

Find the inverse of each of the following exponential functions.

1. $y=3e^{x+2}$
2. $f(x)=\frac{1}{5}e^\frac{x}{7}$
3. $y=2+e^{2x-3}$
4. $f(x)=7^{\frac{3}{x}+1-5}$
5. $y=2(6)^\frac{x-5}{2}$
6. $f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}$

Vocabulary Language: English

Inverse Properties of Logarithms

Inverse Properties of Logarithms

The inverse properties of logarithms are $\log_b b^x=x$ and $b^{\log_b x}=x, b \ne 1$.