8.6: 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
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 of a Function concept), they equal
These are called the Inverse Properties of Logarithms.
Example A
Find:
a)
b)
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.
b) Here,
Example B
Find
Solution: We will use the second property here. Also, rewrite 16 as
Example C
Find the inverse of
Solution: See the Finding the Inverse concept for the steps on how to find the inverse.
Change
Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.
Recall the Inverse Properties from earlier in this concept.
Therefore,
Intro Problem Revisit Switch x and y in the function
Therefore, the inverse of
Guided Practice
1. Simplify
2. Simplify
3. Find the inverse of
Answers
1. Using the first inverse property, the log and the base cancel out, leaving
2. Using the second inverse property and changing 81 into
3. Follow the steps from Example C to find the inverse.
Vocabulary
 Inverse Properties of Logarithms

logbbx=x andblogbx=x,b≠1
Practice
Use the Inverse Properties of Logarithms to simplify the following expressions.

log327x 
log5(15)x 
log2(132)x 
10log(x+3) 
log636(x−1) 
9log9(3x) 
eln(x−7) 
log(1100)3x  \begin{align*}\ln e^{(5x3)}\end{align*}
Find the inverse of each of the following exponential functions.
 \begin{align*}y=3e^{x+2}\end{align*}
 \begin{align*}f(x)=\frac{1}{5}e^\frac{x}{7}\end{align*}
 \begin{align*}y=2+e^{2x3}\end{align*}
 \begin{align*}f(x)=7^{\frac{3}{x}+15}\end{align*}
 \begin{align*}y=2(6)^\frac{x5}{2}\end{align*}
 \begin{align*}f(x)=\frac{1}{3}(8)^{\frac{x}{2}5}\end{align*}
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Here you'll understand the inverse properties of a logarithmic function.