Question

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

Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms states that for any positive numbers \( a \), \( b \), and \( x \) (where \( a
eq 1 \) and \( b
eq 1 \)), \( \log_a x=\frac{\log_b x}{\log_b a} \). This formula allows us to convert a logarithm with base \( a \) to a logarithm with base \( b \) by taking the ratio of the logarithm of \( x \) with base \( b \) to the logarithm of \( a \) with base \( b \).

Step2: Analyze Each Option

• Option 1: \( \frac{\log_a b}{\log_x b} \) does not match the change of base formula. Using the change of base formula, this would be \( \log_x a \) (by applying the formula to \( \log_a b \) and \( \log_x b \)), not \( \log_a x \).
• Option 2: \( \frac{\log_a x}{\log_b x} \) is equal to \( \log_b a \) (by the change of base formula in reverse), not \( \log_a x \).
• Option 3: \( \frac{\log_b x}{\log_b a} \) matches the change of base formula \( \log_a x=\frac{\log_b x}{\log_b a} \).
• Option 4: \( \frac{\log_b a}{\log_b x} \) is the reciprocal of the correct expression, so it is equal to \( \frac{1}{\log_a x} \), not \( \log_a x \).

Answer:

\( \boldsymbol{\frac{\log_b x}{\log_b a}} \) (the third option)