Question

tyler applied the change of base formula to a logarithmic expression. the resulting expression is shown below.\\(\frac{\log \frac{1}{4}}{\log 12}\\)\
which expression could be tylers original expression?\
\\(\circ\\ \log_{\frac{1}{4}} 12\\)\
\\(\circ\\ \log_{12} \frac{1}{4}\\)\
\\(\circ\\ 12\log \frac{1}{4}\\)\
\\(\circ\\ \frac{1}{4}\log 12\\)

Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms is $\log_b a = \frac{\log_c a}{\log_c b}$ for any positive $c
eq 1$. Here, the resulting expression is $\frac{\log \frac{1}{4}}{\log 12}$.

Step2: Match with Original Logarithm

Using the change of base formula, if we have $\log_{12} \frac{1}{4}$, applying the formula (with base $c = 10$ or any common base) gives $\frac{\log \frac{1}{4}}{\log 12}$, which matches the given expression. Let's check other options:

• For $\log_{\frac{1}{4}} 12$, the change of base would be $\frac{\log 12}{\log \frac{1}{4}}$, not the given form.
• $12\log \frac{1}{4}$ is a coefficient multiplication, not related to change of base.
• $\frac{1}{4}\log 12$ is also a coefficient multiplication, not change of base.

Answer:

$\boldsymbol{\log_{12} \frac{1}{4}}$ (corresponding to the option "log₁₂ (1/4)")