To find an expression equivalent to [tex]\(\log_5\left(\frac{x}{4}\right)^2\)[/tex], we can use the properties of logarithms step-by-step.
First, let's rewrite the expression [tex]\(\log_5\left(\frac{x}{4}\right)^2\)[/tex]:
[tex]\[ \log_5\left(\left(\frac{x}{4}\right)^2\right) \][/tex]
Using the logarithm power rule:
[tex]\[ \log_b(a^c) = c \log_b(a) \][/tex]
we can bring the exponent down as a coefficient:
[tex]\[ \log_5\left(\left(\frac{x}{4}\right)^2\right) = 2 \log_5\left(\frac{x}{4}\right) \][/tex]
Next, we use the logarithm quotient rule:
[tex]\[ \log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c) \][/tex]
to split the logarithm:
[tex]\[ 2 \log_5\left(\frac{x}{4}\right) = 2 \left(\log_5(x) - \log_5(4)\right) \][/tex]
Distributing the 2 across the terms inside the parentheses:
[tex]\[ 2 \left(\log_5(x) - \log_5(4)\right) = 2 \log_5(x) - 2 \log_5(4) \][/tex]
Therefore, the expression equivalent to [tex]\(\log_5\left(\frac{x}{4}\right)^2\)[/tex] is:
[tex]\[ 2 \log_5(x) - 2 \log_5(4) \][/tex]
So the correct answer is:
[tex]\[ 2 \log_5 x - 2 \log_5 4 \][/tex]
