Question

for the function, find $f^{-1}$.\\(f(x) = \log_{15}(x)\\)\\(f^{-1}(x) = \square\\)

Explanation:

Step1: Let \( y = f(x) \)

We start by setting \( y = \log_{15}(x) \). This is the original function, and we want to find its inverse, so we need to solve for \( x \) in terms of \( y \).

Step2: Convert to exponential form

Recall that the logarithmic equation \( \log_b(a) = c \) is equivalent to the exponential equation \( b^c = a \). Applying this to \( y = \log_{15}(x) \), we get \( 15^y = x \).

Step3: Swap \( x \) and \( y \)

To find the inverse function, we swap the roles of \( x \) and \( y \). So we replace \( x \) with \( y \) and \( y \) with \( x \) in the equation \( 15^y = x \). This gives us \( 15^x = y \).

Step4: Write the inverse function

The inverse function \( f^{-1}(x) \) is the function we just found, so \( f^{-1}(x) = 15^x \).

Answer:

\( f^{-1}(x) = 15^x \)