Question

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

Explanation:

Step1: Recall the method to find the inverse of a function

To find the inverse of a function \( y = f(x) \), we follow these steps:

1. Replace \( y \) with \( x \) and \( x \) with \( y \) in the equation.
2. Solve the resulting equation for \( y \).

For the exponential function \( y = 6^{x} \), we start by interchanging \( x \) and \( y \):

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

After interchanging \( x \) and \( y \) in \( y = 6^{x} \), we get:
\( x = 6^{y} \)

Step3: Solve for \( y \) using the definition of logarithms

The logarithmic form of the exponential equation \( a^{b}=c \) is \( \log_{a}c = b \).
In our equation \( x = 6^{y} \), we have \( a = 6 \), \( b = y \), and \( c = x \).
Using the definition of logarithms, we can rewrite \( x = 6^{y} \) as:
\( y=\log_{6}x \)

So the inverse of \( y = 6^{x} \) is \( y=\log_{6}x \).

Answer:

A. \( y = \log_{6}x \)