Write as a Single Logarithm log base 4 of x^2-4-6 log base 4 of x+2

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

Simplify $- 6 \log_{4} (x + 2)$ by moving $6$ inside the logarithm.

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

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

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

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Cancel the common factors.

Factor $x + 2$ out of ${(x + 2)}^{6}$ .

$\log_{4} (\frac{(x + 2) (x - 2)}{(x + 2) {(x + 2)}^{5}})$

Cancel the common factor.

$\log_{4} (\frac{(x + 2) (x - 2)}{(x + 2) {(x + 2)}^{5}})$

Rewrite the expression.

$\log_{4} (\frac{x - 2}{{(x + 2)}^{5}})$