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Chapter 3: Problem 59

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\log _{8} x $$

Short Answer

Expert verified
The inverse function of \(f(x)=\log_8 x\) is given by \(f^{-1}(x)=8^x\).

Step by step solution

01

Replace f(x) with y

End up with the equation \(y=\log_8 x\).
02

Swap x and y

Replace x with y and y with x, resulting in the equation \(x=\log_8 y\).
03

Solve for y

To isolate y, we first rewrite the logarithmic equation as an exponential equation. In general, for any logarithm equation with a base b, we can rewrite it as an exponential equation as follows: \(\log_b a = c\) is equivalent to \(b^c = a\). Using this conversion, rewrite the swapped equation as \(8^x = y\).
04

Write the inverse function

Now that the equation is solved for y, we can write the inverse function as \(f^{-1}(x) = 8^x\).

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