45. Which is the inverse of \( y=\log _{5}(x+1) \) ? \( \begin{array}{ll}\text { A. } y=\log _{x+1} 5 & \text { C. } y=5^{x+1} \\ \text { B. } y=5^{x-1} & \text { D. } y=5^{x}-1\end{array} \)

To find the inverse of the function \( y = \log_{5}(x+1) \), we start by switching the roles of \( x \) and \( y \). This gives us \( x = \log_{5}(y+1) \). To isolate \( y \), we exponentiate both sides with base 5, yielding \( 5^x = y + 1 \). Finally, subtracting 1 from both sides leads to \( y = 5^x - 1 \). So, the correct answer is \( D. \) Now, if you're curious about inverses, they can often transform outputs back to their original inputs, creating a nice symmetry! Also, keep in mind that finding inverses requires that the original function be one-to-one (bijective), meaning it should pass the horizontal line test to ensure it doesn’t have duplicate y-values for different x-values.