Evaluate log base 4 of 1/4x^2

$\log_{4} (\frac{1}{4} x^{2})$

Combine $\frac{1}{4}$ and $x^{2}$ .

$\log_{4} (\frac{x^{2}}{4})$

Rewrite $\frac{x^{2}}{4}$ as $x^{2} \cdot 4^{- 1}$ .

$\log_{4} (x^{2} \cdot 4^{- 1})$

Rewrite $\log_{4} (x^{2} \cdot 4^{- 1})$ as $\log_{4} (x^{2}) + \log_{4} (4^{- 1})$ .

$\log_{4} (x^{2}) + \log_{4} (4^{- 1})$

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{4} (x^{2}) - \log_{4} (4)$

Logarithm base $4$ of $4$ is $1$ .

$\log_{4} (x^{2}) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$\log_{4} (x^{2}) - 1$

Rewrite $\log_{4} (x^{2})$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (x^{2})}{\log (4)} - 1$