Question

what is the inverse of the logarithmic function $f(x) = \log_{9}x$?
$\circ$ $f^{-1}(x) = x^{9}$ $\circ$ $f^{-1}(x) = -\log_{9}x$
$\circ$ $f^{-1}(x) = 9^{x}$ $\circ$ $f^{-1}(x) = \frac{1}{\log_{9} x}$

Explanation:

Step1: Recall the definition of inverse function

To find the inverse of a function \( y = f(x) \), we first swap \( x \) and \( y \), then solve for \( y \). For the logarithmic function \( y=\log_{9}x \), we know that the logarithmic function \( y = \log_{a}x \) (where \( a>0,a
eq1 \)) and the exponential function \( y = a^{x} \) are inverse functions of each other.

Step2: Swap \( x \) and \( y \)

Start with \( y=\log_{9}x \). Swap \( x \) and \( y \) to get \( x = \log_{9}y \).

Step3: Convert logarithmic form to exponential form

Using the definition of logarithms, if \( x=\log_{b}a \), then \( a = b^{x} \). So for \( x=\log_{9}y \), we can rewrite it in exponential form as \( y = 9^{x} \).

Step4: Write the inverse function

So the inverse function \( f^{-1}(x) \) is \( f^{-1}(x)=9^{x} \).

Answer:

\( f^{-1}(x) = 9^{x} \) (corresponding to the option \( f^{-1}(x) = 9^{x} \))