About Inverse of an Exponential / Logarithmic Function:

In some cases, we will need to find the inverse of an exponential function: f(x) = ax, where a is greater than zero and a is not equal to 1. Additionally, we may need to find the inverse of a logarithmic function: f(x) = loga(x), where a is greater than zero, a is not equal to 1, and x is greater than zero.


Test Objectives
  • Demonstrate the ability to find the inverse of an exponential function
  • Demonstrate the ability to find the inverse of a logarithmic function
Inverse of an Exponential / Logarithmic Function Practice Test:

#1:

Instructions: find the inverse.

$$a)\hspace{.2em}f(x)=log_{3}(x^3 + 2) + 7$$

$$b)\hspace{.2em}f(x)=7log_{2}(x - 3) - 3$$


#2:

Instructions: find the inverse.

$$a)\hspace{.2em}f(x)=log_{3}(-3x - 6) - 4$$

$$b)\hspace{.2em}f(x)=\left(\frac{4^x + 4}{4}\right)^{\frac{1}{4}}$$


#3:

Instructions: find the inverse.

$$a)\hspace{.2em}f(x)=\left(\frac{4^x - 2}{-4}\right)^{\frac{1}{4}}$$

$$b)\hspace{.2em}f(x)=\frac{-1}{2^{x + 2}}$$


#4:

Instructions: find the inverse.

$$a)\hspace{.2em}f(x)=4^x + 4$$

$$b)\hspace{.2em}f(x)=\frac{6^x}{2}$$


#5:

Instructions: find the inverse.

$$a)\hspace{.2em}f(x)=\frac{1}{\sqrt[4]{3^x}}$$

$$b)\hspace{.2em}f(x)=\frac{4 \cdot 3^x + 1}{3^x}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=\sqrt[3]{3^{x - 7}- 2}$$

$$b)\hspace{.2em}f^{-1}(x)=2^{\frac{x + 3}{7}}+ 3$$


#2:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=\frac{3^{x + 4}+ 6}{-3}$$

$$b)\hspace{.2em}f^{-1}(x)=log_{4}(4x^4 - 4)$$


#3:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=log_{4}(-4x^4 + 2)$$

$$b)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{2}}(-4x)$$


#4:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=log_{4}(x - 4)$$

$$b)\hspace{.2em}f^{-1}(x)=log_{6}(2x)$$


#5:

Solutions:

$$a)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{3}}(x^4)$$

$$b)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{3}}(x - 4)$$