Question

which of the following is the inverse of $y = 6^x$?
$\bigcirc\\ y = \log_{6} x$
$\bigcirc\\ y = \log_{x} 6$
$\bigcirc\\ y = \log_{\frac{1}{6}} x$
$\bigcirc\\ y = \log_{6} 6x$

Explanation:

Step1: Recall the inverse of exponential function

For an exponential function \( y = a^x \) (where \( a>0, a
eq1 \)), its inverse function is the logarithmic function \( y=\log_{a}x \) by the definition of inverse functions (swap \( x \) and \( y \) and solve for \( y \)).

Step2: Apply the rule to \( y = 6^x \)

Given the function \( y = 6^x \), to find its inverse, we first swap \( x \) and \( y \), getting \( x = 6^y \). Then we convert this exponential equation to a logarithmic equation. By the definition of logarithms, if \( x = a^y \), then \( y=\log_{a}x \). So for \( x = 6^y \), we have \( y=\log_{6}x \).

Answer:

\( y = \log_{6}x \) (the first option)