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.
$$a)\hspace{.2em}f(x)=\text{log}_{3}(x^3 + 2) + 7$$
$$b)\hspace{.2em}f(x)=7\text{log}_{2}(x - 3) - 3$$
Watch the Step by Step Video Solution View the Written Solution
#2:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=\text{log}_{3}(-3x - 6) - 4$$
$$b)\hspace{.2em}f(x)=\left(\frac{4^x + 4}{4}\right)^{\frac{1}{4}}$$ For this problem, you need to consider the range. This will become the domain in the inverse. $$\text{Range}:\{y | y > 1\}$$
Watch the Step by Step Video Solution View the Written Solution
#3:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=\left(\frac{4^x - 2}{-4}\right)^{\frac{1}{4}}$$ For this problem, you need to consider the range. This will become the domain in the inverse. $$\text{Range}:\left\{y | 0 ≤ y < \frac{\sqrt[4]{8}}{2}\right\}$$
$$b)\hspace{.2em}f(x)=\frac{-1}{2^{x + 2}}$$
Watch the Step by Step Video Solution View the Written Solution
#4:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=4^x + 4$$
$$b)\hspace{.2em}f(x)=\frac{6^x}{2}$$
Watch the Step by Step Video Solution View the Written Solution
#5:
Instructions: find the inverse.
$$a)\hspace{.2em}f(x)=\frac{1}{\sqrt[4]{3^x}}$$ For this problem, you need to consider the range. This will become the domain in the inverse. $$\text{Range}:\{y | y > 0\}$$
$$b)\hspace{.2em}f(x)=\frac{4 \cdot 3^x + 1}{3^x}$$
Watch the Step by Step Video Solution View the Written Solution
Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\sqrt[3]{3^{x - 7}- 2}$$
Graph Key:
$$f^{-1}(x)=\sqrt[3]{3^{x - 7}- 2}$$
$$y=x$$
$$f(x)=\text{log}_{3}(x^3 + 2) + 7$$
$$b)\hspace{.2em}f^{-1}(x)=\large{2^{\frac{x + 3}{7}}}+ 3$$
Graph Key:
$$f^{-1}(x)=\large{2^{\frac{x + 3}{7}}}+ 3$$
$$y=x$$
$$f(x)=7\text{log}_{2}(x - 3) - 3$$
Watch the Step by Step Video Solution
#2:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=-3^{x + 3}- 2$$
Graph Key:
$$f^{-1}(x)=-3^{x + 3}- 2$$
$$y=x$$
$$f(x)=\text{log}_{3}(-3x - 6) - 4$$
$$b)\hspace{.2em}f^{-1}(x)=\text{log}_{4}(4x^4 - 4), x > 1$$
Graph Key:
$$f^{-1}(x)=\text{log}_{4}(4x^4 - 4), x > 1$$
$$y=x$$
$$f(x)=\left(\frac{4^x + 4}{4}\right)^{\frac{1}{4}}$$
Watch the Step by Step Video Solution
#3:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\text{log}_{4}(-4x^4 + 2), 0 ≤ x < \frac{\sqrt[4]{8}}{2}$$
Graph Key:
$$f^{-1}(x)=\text{log}_{4}(-4x^4 + 2), 0 ≤ x < \frac{\sqrt[4]{8}}{2}$$
$$y=x$$
$$f(x)=\left(\frac{4^x - 2}{-4}\right)^{\frac{1}{4}}$$
$$b)\hspace{.2em}f^{-1}(x)=\text{log}_{\frac{1}{2}}(-4x)$$
Graph Key:
$$f^{-1}(x)=\text{log}_{\frac{1}{2}}(-4x)$$
$$y=x$$
$$f(x)=\frac{-1}{2^{x + 2}}$$
Watch the Step by Step Video Solution
#4:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\text{log}_{4}(x - 4)$$
Graph Key:
$$f^{-1}(x)=\text{log}_{4}(x - 4)$$
$$y=x$$
$$f(x)=4^x + 4$$
$$b)\hspace{.2em}f^{-1}(x)=\text{log}_{6}(2x)$$
Graph Key:
$$f^{-1}(x)=\text{log}_{6}(2x)$$
$$y=x$$
$$f(x)=\frac{6^x}{2}$$
Watch the Step by Step Video Solution
#5:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\text{log}_{\frac{1}{3}}(x^4), x > 0$$
Graph Key:
$$f^{-1}(x)=\text{log}_{\frac{1}{3}}(x^4), x > 0$$
$$y=x$$
$$f(x)=\frac{1}{\sqrt[4]{3^x}}$$
$$b)\hspace{.2em}f^{-1}(x)=\text{log}_{\frac{1}{3}}(x - 4)$$
Graph Key:
$$f^{-1}(x)=\text{log}_{\frac{1}{3}}(x - 4)$$
$$y=x$$
$$f(x)=\frac{4 \cdot 3^x + 1}{3^x}$$
