Lesson Objectives
- Learn how to find the Inverse of an Exponential Function
- Learn how to find the Inverse of a Logarithmic Function
How to Find the Inverse of an Exponential or Logarithmic Function
In this lesson, we want to learn how to find the inverse of an exponential or logarithmic function. Before we get into this topic, we want to review two properties for logarithms and how they relate to function composition. If two functions let’s say f and g are inverses, then: $$f(g(x))=x$$ and $$g(f(x))=x$$ Now, let's think about what happens when we use function composition with an exponential and logarithmic function. $$f(x)=a^x$$ $$g(x)=log_a(x)$$ Restrictions: $$a > 0, a ≠ 1, x > 0$$ $$f(g(x))=a^{log_{a}(x)}=x$$ We have already learned this as a special property of logarithms: $$a^{log_a(x)}=x$$ Now, let's go the other way: $$g(f(x))=log_a(a^x)=x$$ $$xlog_a(a)=x \cdot 1=x$$ Again, we already learned that: $$log_a(a)=1$$ In general: $$log_a(a^x)=x$$ Let's use these tools to find the inverse of an exponential function.
Example #1: Find the inverse of each. $$f(x)=log_3(2^x)$$ Step 1) Replace f(x) with y: $$y=log_3(2^x)$$ Step 2) Interchange x and y: $$x=log_3(2^y)$$ Step 3) Solve for y: $$3^x=2^y$$ $$log_{2}(3^x)=log_2(2^y)$$ $$log_{2}(3^x)=y$$ Step 4) Replace y with f-1(x) $$f^{-1}(x)=log_2(3^x)$$
Example #1: Find the inverse of each. $$f(x)=log_3(2^x)$$ Step 1) Replace f(x) with y: $$y=log_3(2^x)$$ Step 2) Interchange x and y: $$x=log_3(2^y)$$ Step 3) Solve for y: $$3^x=2^y$$ $$log_{2}(3^x)=log_2(2^y)$$ $$log_{2}(3^x)=y$$ Step 4) Replace y with f-1(x) $$f^{-1}(x)=log_2(3^x)$$
Skills Check:
Example #1
Find the inverse. $$\large{f(x)=3^{\frac{x}{2}}}$$
Please choose the best answer.
A
$$f^{-1}(x)=log_{3}(4x)$$
B
$$f^{-1}(x)=log_{3}(x^2)$$
C
$$f^{-1}(x)=log_{4}(-3x)$$
D
$$f^{-1}(x)=log_{4}(x - 9)$$
E
$$f^{-1}(x)=log_{4}(x - 1)$$
Example #2
Find the inverse. $$\large{f(x)=log_{\frac{1}{2}}(-4x)}$$
Please choose the best answer.
A
$$f^{-1}(x)=2^x + 1$$
B
$$f^{-1}(x)=6^x + 10$$
C
$$f^{-1}(x)=\frac{5^x}{4}$$
D
$$f^{-1}(x)=-\frac{1}{2^{x + 2}}$$
E
$$f^{-1}(x)=4^{-\frac{4}{x}}$$
Example #3
Find the inverse. $$f(x)=\frac{8 \cdot 3^x + 1}{3^x}$$
Please choose the best answer.
A
$$f^{-1}(x)=log_{5}(2x)$$
B
$$f^{-1}(x)=log_{5}(x + 7)$$
C
$$f^{-1}(x)=log_{3}(x - 3)$$
D
$$f^{-1}(x)=log_{\frac{1}{3}}(x - 8)$$
E
$$f^{-1}(x)=log_4(x - 1)$$
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