Precalculus Examples

$\log_{4} (\frac{x^{6} y^{5}}{z \sqrt[5]{w^{6}}})$

Step 1

Rewrite $\log_{4} (\frac{x^{6} y^{5}}{z \sqrt[5]{w^{6}}})$ as $\log_{4} (x^{6} y^{5}) - \log_{4} (z \sqrt[5]{w^{6}})$ .

$\log_{4} (x^{6} y^{5}) - \log_{4} (z \sqrt[5]{w^{6}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{w^{6}}$ as $w^{\frac{6}{5}}$ .

$\log_{4} (x^{6} y^{5}) - \log_{4} (z w^{\frac{6}{5}})$

Step 3

Rewrite $\log_{4} (x^{6} y^{5})$ as $\log_{4} (x^{6}) + \log_{4} (y^{5})$ .

$\log_{4} (x^{6}) + \log_{4} (y^{5}) - \log_{4} (z w^{\frac{6}{5}})$

Step 4

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

$6 \log_{4} (x) + \log_{4} (y^{5}) - \log_{4} (z w^{\frac{6}{5}})$

Step 5

Expand $\log_{4} (y^{5})$ by moving $5$ outside the logarithm.

$6 \log_{4} (x) + 5 \log_{4} (y) - \log_{4} (z w^{\frac{6}{5}})$

Step 6

Rewrite $\log_{4} (z w^{\frac{6}{5}})$ as $\log_{4} (z) + \log_{4} (w^{\frac{6}{5}})$ .

$6 \log_{4} (x) + 5 \log_{4} (y) - (\log_{4} (z) + \log_{4} (w^{\frac{6}{5}}))$

Step 7

Expand $\log_{4} (w^{\frac{6}{5}})$ by moving $\frac{6}{5}$ outside the logarithm.

$6 \log_{4} (x) + 5 \log_{4} (y) - (\log_{4} (z) + \frac{6}{5} \log_{4} (w))$

Step 8

Simplify each term.

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Step 8.1

Combine $\frac{6}{5}$ and $\log_{4} (w)$ .

$6 \log_{4} (x) + 5 \log_{4} (y) - (\log_{4} (z) + \frac{6 \log_{4} (w)}{5})$

Step 8.2

Apply the distributive property.

$6 \log_{4} (x) + 5 \log_{4} (y) - \log_{4} (z) - \frac{6 \log_{4} (w)}{5}$