3.3.3: Inverse Properties of Logarithms
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 14374
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?
Inverse Properties of Logarithms
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, they equal \(\ x\). Therefore, if \(\ f(x)=b^{x} \text { and } g(x)=\log _{b} x\), then:
\(\ f \circ g=b^{\log _{b} x}=x \text { and } g \circ f=\log _{b} b^{x}=x\)
These are called the Inverse Properties of Logarithms.
Let's solve the following problems. We will use the Inverse Properties of Logarithms.

Find \(\ 10^{\log 56}\).
Using the first property, we see that the bases cancel each other out. \(\ 10^{\log 56}=56\)
\(\ e^{\ln 6} \cdot e^{\ln 2}\)
Here, \(\ e\) and the natural log cancel out and we are left with 6⋅2=12.

Find \(\ \log _{4} 16^{x}\)
We will use the second property here. Also, rewrite 16 as 4^{2}.\(\ \log _{4} 16^{x}=\log _{4}\left(4^{2}\right)^{x}=\log _{4} 4^{2 x}=2 x\)

Find the inverse of \(\ f(x)=2 e^{x1}\).
Change \(\ f(x)\) to \(\ y\). Then, switch \(\ x\) and \(\ y\).
\(\ \begin{array}{l}
y=2 e^{x1} \\
x=2 e^{y1}
\end{array}\)Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.
\(\ \begin{array}{l}
\frac{x}{2}=e^{y1} \\
\ln \left(\frac{x}{2}\right)=\ln e^{y1}
\end{array}\)Recall the Inverse Properties of Logarithms from earlier in this concept. \(\ \log _{b} b^{x}=x\); applying this to the right side of our equation, we have \(\ \ln e^{y1}=y1\). Solve for \(\ y\).
\(\ \begin{array}{l}
\ln \left(\frac{x}{2}\right)=y1 \\
\ln \left(\frac{x}{2}\right)+1=y
\end{array}\)Therefore, \(\ \ln \left(\frac{x}{2}\right)+1\) is the inverse of \(\ 2 e^{y1}\).
Examples
Example 1
Earlier, you were asked to find the inverse of \(\ y=C e^{x}\).
Solution
Switch x and y in the function \(\ y=C e^{x}\) and then solve for y.
\(\ \begin{array}{r}
x=C e^{y} \\
\frac{x}{C}=e^{y} \\
\ln \frac{x}{C}=\ln \left(e^{y}\right) \\
\ln \frac{x}{C}=y
\end{array}\)
Therefore, the inverse of \(\ y=C e^{x} \text { is } y=\ln \frac{x}{C}\).
Example 2
Simplify \(\ 5^{\log _{5} 6 x}\).
Solution
Using the first inverse property, the log and the base cancel out, leaving \(\ 6x\) as the answer.
\(\ 5^{\log _{5} 6 x}=6 x\)
Example 3
Simplify \(\ \log _{9} 81^{x+2}\).
Solution
Using the second inverse property and changing 81 into 9^{2} we have:
\(\ \begin{aligned}
\log _{9} 81^{x+2} &=\log _{9} 9^{2(x+2)} \\
&=2(x+2) \\
&=2 x+4
\end{aligned}\)
Example 4
Find the inverse of \(\ f(x)=4^{x+2}5\).
Solution
\(\ \begin{aligned}
f(x) &=4^{x+2}5 \\
y &=4^{x+2}5 \\
x &=4^{y+2}5 \\
x+5 &=4^{y+2} \\
\log _{4}(x+5) &=y+2 \\
\log _{4}(x+5)2 &=y
\end{aligned}\)
Review
Use the Inverse Properties of Logarithms to simplify the following expressions.
 \(\ \log _{3} 27^{x}\)
 \(\ \log _{5}\left(\frac{1}{5}\right)^{x}\)
 \(\ \log _{2}\left(\frac{1}{32}\right)^{x}\)
 \(\ 10^{\log (x+3)}\)
 \(\ \log _{6} 36^{(x1)}\)
 \(\ 9^{\log _{9}(3 x)}\)
 \(\ e^{\ln (x7)}\)
 \(\ \log \left(\frac{1}{100}\right)^{3 x}\)
 \(\ \ln e^{(5 x3)}\)
Find the inverse of each of the following exponential functions.
 \(\ y=3 e^{x+2}\)
 \(\ f(x)=\frac{1}{5} e^{\frac{x}{7}}\)
 \(\ y=2+e^{2 x3}\)
 \(\ f(x)=7^{\frac{3}{x}+15}\)
 \(\ y=2(6)^{\frac{x5}{2}}\)
 \(\ f(x)=\frac{1}{3}(8)^{\frac{x}{2}5}\)
Vocabulary
Term  Definition 

Inverse Properties of Logarithms  The inverse properties of logarithms are \(\ \log _{b} b^{x}=x \text { and } b^{\log _{b} x}=x, b \neq 1\). 