Expand the Logarithmic Expression log base 3 of (4x)/((7y)^9)

$\log_{3} (\frac{4 x}{{(7 y)}^{9}})$

Step 1

Rewrite $\log_{3} (\frac{4 x}{{(7 y)}^{9}})$ as $\log_{3} (4 x) - \log_{3} ({(7 y)}^{9})$ .

$\log_{3} (4 x) - \log_{3} ({(7 y)}^{9})$

Step 2

Expand $\log_{3} ({(7 y)}^{9})$ by moving $9$ outside the logarithm.

$\log_{3} (4 x) - (9 \log_{3} (7 y))$

Step 3

Multiply $9$ by $- 1$ .

$\log_{3} (4 x) - 9 \log_{3} (7 y)$

Step 4

Rewrite $\log_{3} (4 x)$ as $\log_{3} (4) + \log_{3} (x)$ .

$\log_{3} (4) + \log_{3} (x) - 9 \log_{3} (7 y)$

Step 5

Rewrite $\log_{3} (7 y)$ as $\log_{3} (7) + \log_{3} (y)$ .

$\log_{3} (4) + \log_{3} (x) - 9 (\log_{3} (7) + \log_{3} (y))$

Step 6

Simplify each term.

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

$\log_{3} (2^{2}) + \log_{3} (x) - 9 (\log_{3} (7) + \log_{3} (y))$

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

$2 \log_{3} (2) + \log_{3} (x) - 9 (\log_{3} (7) + \log_{3} (y))$

Apply the distributive property.

$2 \log_{3} (2) + \log_{3} (x) - 9 \log_{3} (7) - 9 \log_{3} (y)$