Expand the Logarithmic Expression log base 7 of (11/(10^2))^4

$\log_{7} ({(\frac{11}{10^{2}})}^{4})$

Step 1

Expand $\log_{7} ({(\frac{11}{10^{2}})}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{7} (\frac{11}{10^{2}})$

Step 2

Raise $10$ to the power of $2$ .

$4 \log_{7} (\frac{11}{100})$

Step 3

Rewrite $\log_{7} (\frac{11}{100})$ as $\log_{7} (11) - \log_{7} (100)$ .

$4 (\log_{7} (11) - \log_{7} (100))$

Step 4

Rewrite $\log_{7} (100)$ as $\log_{7} (2^{2} \cdot 5^{2})$ .

$4 (\log_{7} (11) - \log_{7} (2^{2} \cdot 5^{2}))$

Step 5

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

$4 (\log_{7} (11) - (\log_{7} (2^{2}) + \log_{7} (5^{2})))$

Step 6

Simplify each term.

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

$4 (\log_{7} (11) - (2 \log_{7} (2) + \log_{7} (5^{2})))$

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

$4 (\log_{7} (11) - (2 \log_{7} (2) + 2 \log_{7} (5)))$

Step 7

Apply the distributive property.

$4 (\log_{7} (11) - (2 \log_{7} (2)) - (2 \log_{7} (5)))$

Step 8

Multiply $2$ by $- 1$ .

$4 (\log_{7} (11) - 2 \log_{7} (2) - (2 \log_{7} (5)))$

Step 9

Multiply $2$ by $- 1$ .

$4 (\log_{7} (11) - 2 \log_{7} (2) - 2 \log_{7} (5))$

Step 10

Apply the distributive property.

$4 \log_{7} (11) + 4 (- 2 \log_{7} (2)) + 4 (- 2 \log_{7} (5))$

Step 11

Simplify.

Multiply $- 2$ by $4$ .

$4 \log_{7} (11) - 8 \log_{7} (2) + 4 (- 2 \log_{7} (5))$

Multiply $- 2$ by $4$ .

$4 \log_{7} (11) - 8 \log_{7} (2) - 8 \log_{7} (5)$