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 tyler’s 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 $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive number not equal to 1.

Step2: Analyze the Given Expression

The resulting expression is $\frac{\log \frac{1}{4}}{\log 12}$. Comparing with the change of base formula $\frac{\log_c a}{\log_c b}$, we can see that $a = \frac{1}{4}$ and $b = 12$ (assuming the common logarithm base, i.e., $c = 10$ or natural logarithm base, but the base in the formula just needs to be consistent). So the original logarithmic expression should be $\log_{12} \frac{1}{4}$, because using the change of base formula on $\log_{12} \frac{1}{4}$ gives $\frac{\log \frac{1}{4}}{\log 12}$.

Answer:

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