Question

which expression can be used to approximate the expression below, for all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1?
$log_{a}x$
$\bigcirc$ $\frac{log_{b}x}{log_{b}a}$
$\bigcirc$ $\frac{log_{b}a}{log_{b}x}$
$\bigcirc$ $\frac{log_{b}b}{log_{x}b}$
$\bigcirc$ $\frac{log_{a}x}{log_{b}x}$

Explanation:

Step1: Recall change of base formula

The change of base formula for logarithms states that for positive numbers $a$, $b$, $x$ (where $a
eq 1$, $b
eq 1$):
$$\log_a x = \frac{\log_b x}{\log_b a}$$

Step2: Match with given options

Compare the formula to the provided choices to identify the correct equivalent expression.

Answer:

$\boldsymbol{\frac{\log_b x}{\log_b a}}$