Precalculus Examples

$\log_{4} (\frac{x^{6}}{\frac{9 (y^{4})}{9}})$

Rewrite $\log_{4} (\frac{x^{6}}{\frac{9 (y^{4})}{9}})$ as $\log_{4} (x^{6}) - \log_{4} (\frac{9 (y^{4})}{9})$ .

$\log_{4} (x^{6}) - \log_{4} (\frac{9 (y^{4})}{9})$

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

$\log_{4} (x^{6}) - (\log_{4} (9 (y^{4})) - \log_{4} (9))$

Expand $\log_{4} (x^{6})$ by moving $6$ outside the logarithm.

$6 \log_{4} (x) - (\log_{4} (9 (y^{4})) - \log_{4} (9))$

Rewrite $\log_{4} (9)$ as $\log_{4} (3^{2})$ .

$6 \log_{4} (x) - (\log_{4} (9 (y^{4})) - \log_{4} (3^{2}))$

Expand $\log_{4} (3^{2})$ by moving $2$ outside the logarithm.

$6 \log_{4} (x) - (\log_{4} (9 (y^{4})) - (2 \log_{4} (3)))$

Multiply $2$ by $- 1$ .

$6 \log_{4} (x) - (\log_{4} (9 (y^{4})) - 2 \log_{4} (3))$

Apply the distributive property.

$6 \log_{4} (x) - \log_{4} (9 y^{4}) - (- 2 \log_{4} (3))$

Multiply $- 2$ by $- 1$ .

$6 \log_{4} (x) - \log_{4} (9 y^{4}) + 2 \log_{4} (3)$

Rewrite $\log_{4} (9 y^{4})$ as $\log_{4} (9) + \log_{4} (y^{4})$ .

$6 \log_{4} (x) - (\log_{4} (9) + \log_{4} (y^{4})) + 2 \log_{4} (3)$

Expand $\log_{4} (y^{4})$ by moving $4$ outside the logarithm.

$6 \log_{4} (x) - (\log_{4} (9) + 4 \log_{4} (y)) + 2 \log_{4} (3)$

Simplify each term.

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Simplify each term.

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Rewrite $\log_{4} (9)$ as $\log_{4} (3^{2})$ .

$6 \log_{4} (x) - (\log_{4} (3^{2}) + 4 \log_{4} (y)) + 2 \log_{4} (3)$

Expand $\log_{4} (3^{2})$ by moving $2$ outside the logarithm.

$6 \log_{4} (x) - (2 \log_{4} (3) + 4 \log_{4} (y)) + 2 \log_{4} (3)$

Apply the distributive property.

$6 \log_{4} (x) - (2 \log_{4} (3)) - (4 \log_{4} (y)) + 2 \log_{4} (3)$

Multiply $2$ by $- 1$ .

$6 \log_{4} (x) - 2 \log_{4} (3) - (4 \log_{4} (y)) + 2 \log_{4} (3)$

Multiply $4$ by $- 1$ .

$6 \log_{4} (x) - 2 \log_{4} (3) - 4 \log_{4} (y) + 2 \log_{4} (3)$

Add $- 2 \log_{4} (3)$ and $2 \log_{4} (3)$ .

$6 \log_{4} (x) + 0 - 4 \log_{4} (y)$

Add $6 \log_{4} (x)$ and $0$ .

$6 \log_{4} (x) - 4 \log_{4} (y)$