Precalculus Examples

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

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

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

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

Step 3

Rewrite $\log_{3} (7 y^{9})$ as $\log_{3} (7) + \log_{3} (y^{9})$ .

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $9$ by $- 1$ .

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