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Chapter 3: Problem 60

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\log _{3} x $$

Short Answer

Expert verified
The inverse function of \(f(x) = \log_3{x}\) is \(f^{-1}(x) = 3^x\).

Step by step solution

01

Rewrite the function with y

We start by replacing \(f(x)\) with \(y\), so the function becomes: $$ y = \log_3{x} $$
02

Swap x and y

Now, we swap x and y in the equation, as follows: $$ x = \log_3{y} $$
03

Solve for y

Now we need to solve for y. To do this, we will use the fact that \(\log_a{b} = c\) is equivalent to \(a^c = b\). In our case, this means that: $$ 3^x = y $$
04

Write the inverse function

Now that we have solved for y in terms of x, we can write the inverse function, \(f^{-1}(x)\): $$ f^{-1}(x) = 3^x $$ So the inverse function of \(f(x) = \log_3{x}\) is \(f^{-1}(x) = 3^x\).

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