Simplify/Condense log base 4 of x^2-36-4 log base 4 of x+6

$\log_{4} (x^{2} - 36) - 4 \log_{4} (x + 6)$

Simplify $- 4 \log_{4} (x + 6)$ by moving $4$ inside the logarithm.

$\log_{4} (x^{2} - 36) - \log_{4} ({(x + 6)}^{4})$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{x^{2} - 36}{{(x + 6)}^{4}})$

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\log_{4} (\frac{x^{2} - 6^{2}}{{(x + 6)}^{4}})$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\log_{4} (\frac{(x + 6) (x - 6)}{{(x + 6)}^{4}})$

Cancel the common factors.

Factor $x + 6$ out of ${(x + 6)}^{4}$ .

$\log_{4} (\frac{(x + 6) (x - 6)}{(x + 6) {(x + 6)}^{3}})$

Cancel the common factor.

$\log_{4} (\frac{(x + 6) (x - 6)}{(x + 6) {(x + 6)}^{3}})$

Rewrite the expression.

$\log_{4} (\frac{x - 6}{{(x + 6)}^{3}})$