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
#1:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=log_{3}(x^3 + 2) + 7$$
$$b)\hspace{.2em}f(x)=7log_{2}(x - 3) - 3$$
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#2:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=log_{3}(-3x - 6) - 4$$
$$b)\hspace{.2em}f(x)=\left(\frac{4^x + 4}{4}\right)^{\frac{1}{4}}$$
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#3:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=\left(\frac{4^x - 2}{-4}\right)^{\frac{1}{4}}$$
$$b)\hspace{.2em}f(x)=\frac{-1}{2^{x + 2}}$$
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#4:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=4^x + 4$$
$$b)\hspace{.2em}f(x)=\frac{6^x}{2}$$
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#5:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=\frac{1}{\sqrt[4]{3^x}}$$
$$b)\hspace{.2em}f(x)=\frac{4 \cdot 3^x + 1}{3^x}$$
Watch the Step by Step Video Lesson View the Written Solution
Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\sqrt[3]{3^{x - 7}- 2}$$
$$b)\hspace{.2em}f^{-1}(x)=2^{\frac{x + 3}{7}}+ 3$$
Watch the Step by Step Video Lesson
#2:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\frac{3^{x + 4}+ 6}{-3}$$
$$b)\hspace{.2em}f^{-1}(x)=log_{4}(4x^4 - 4)$$
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#3:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=log_{4}(-4x^4 + 2)$$
$$b)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{2}}(-4x)$$
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#4:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=log_{4}(x - 4)$$
$$b)\hspace{.2em}f^{-1}(x)=log_{6}(2x)$$
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#5:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{3}}(x^4)$$
$$b)\hspace{.2em}f^{-1}(x)=log_{\frac{1}{3}}(x - 4)$$
