If \( f(x)=9^{x} \), what is \( f^{-1}(x) ? \) \( f^{-1}(x)=\square \)

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = 9^x \), we start by setting \( y = f(x) \): \[ y = 9^x \] Next, we want to solve for \( x \) in terms of \( y \). To do this, we can take the logarithm of both sides. We can use the natural logarithm (or any logarithm), but for simplicity, let's use the base 9 logarithm: \[ \log_9(y) = x \] Now, we can express \( x \) in terms of \( y \): \[ x = \log_9(y) \] Since we are looking for the inverse function \( f^{-1}(x) \), we replace \( y \) with \( x \): \[ f^{-1}(x) = \log_9(x) \] To express \( \log_9(x) \) in terms of natural logarithms, we can use the change of base formula: \[ \log_9(x) = \frac{\ln(x)}{\ln(9)} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{\ln(x)}{\ln(9)} \] So, the final answer is: \[ f^{-1}(x) = \frac{\ln(x)}{\ln(9)} \]