Expand the Logarithmic Expression log base 7 of ((5^2)/8)^6

$\log_{7} ({(\frac{5^{2}}{8})}^{6})$

Expand $\log_{7} ({(\frac{5^{2}}{8})}^{6})$ by moving $6$ outside the logarithm.

$6 \log_{7} (\frac{5^{2}}{8})$

Raise $5$ to the power of $2$ .

$6 \log_{7} (\frac{25}{8})$

Rewrite $\log_{7} (\frac{25}{8})$ as $\log_{7} (25) - \log_{7} (8)$ .

$6 (\log_{7} (25) - \log_{7} (8))$

Rewrite $\log_{7} (8)$ as $\log_{7} (2^{3})$ .

$6 (\log_{7} (25) - \log_{7} (2^{3}))$

Expand $\log_{7} (2^{3})$ by moving $3$ outside the logarithm.

$6 (\log_{7} (25) - (3 \log_{7} (2)))$

Multiply $3$ by $- 1$ .

$6 (\log_{7} (25) - 3 \log_{7} (2))$

Simplify each term.

Rewrite $\log_{7} (25)$ as $\log_{7} (5^{2})$ .

$6 (\log_{7} (5^{2}) - 3 \log_{7} (2))$

Expand $\log_{7} (5^{2})$ by moving $2$ outside the logarithm.

$6 (2 \log_{7} (5) - 3 \log_{7} (2))$

Apply the distributive property.

$6 (2 \log_{7} (5)) + 6 (- 3 \log_{7} (2))$

Multiply $2$ by $6$ .

$12 \log_{7} (5) + 6 (- 3 \log_{7} (2))$

Multiply $- 3$ by $6$ .

$12 \log_{7} (5) - 18 \log_{7} (2)$