Write as a Single Logarithm log base 4 of x^2-25-4 log base 4 of x+5

$\log_{4} (x^{2} - 25) - 4 \log_{4} (x + 5)$

Simplify $- 4 \log_{4} (x + 5)$ by moving $4$ inside the logarithm.

$\log_{4} (x^{2} - 25) - \log_{4} ({(x + 5)}^{4})$

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

$\log_{4} (\frac{x^{2} - 25}{{(x + 5)}^{4}})$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

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

Cancel the common factors.

Factor $x + 5$ out of ${(x + 5)}^{4}$ .

$\log_{4} (\frac{(x + 5) (x - 5)}{(x + 5) {(x + 5)}^{3}})$

Cancel the common factor.

$\log_{4} (\frac{(x + 5) (x - 5)}{(x + 5) {(x + 5)}^{3}})$

Rewrite the expression.

$\log_{4} (\frac{x - 5}{{(x + 5)}^{3}})$