Precalculus Examples

$\log_{3} (x^{4} y^{2}) - \log_{3} (z \sqrt[5]{w^{3}})$

Step 1

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

$\log_{3} (x^{4}) + \log_{3} (y^{2}) - \log_{3} (z \sqrt[5]{w^{3}})$

Step 2

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

$4 \log_{3} (x) + \log_{3} (y^{2}) - \log_{3} (z \sqrt[5]{w^{3}})$

Step 3

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

$4 \log_{3} (x) + 2 \log_{3} (y) - \log_{3} (z \sqrt[5]{w^{3}})$

Step 4

Rewrite $\log_{3} (z \sqrt[5]{w^{3}})$ as $\log_{3} (z) + \log_{3} (\sqrt[5]{w^{3}})$ .

$4 \log_{3} (x) + 2 \log_{3} (y) - (\log_{3} (z) + \log_{3} (\sqrt[5]{w^{3}}))$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{w^{3}}$ as $w^{\frac{3}{5}}$ .

$4 \log_{3} (x) + 2 \log_{3} (y) - (\log_{3} (z) + \log_{3} (w^{\frac{3}{5}}))$

Step 6

Expand $\log_{3} (w^{\frac{3}{5}})$ by moving $\frac{3}{5}$ outside the logarithm.

$4 \log_{3} (x) + 2 \log_{3} (y) - (\log_{3} (z) + \frac{3}{5} \log_{3} (w))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{3}{5}$ and $\log_{3} (w)$ .

$4 \log_{3} (x) + 2 \log_{3} (y) - (\log_{3} (z) + \frac{3 \log_{3} (w)}{5})$

Step 7.2

Apply the distributive property.

$4 \log_{3} (x) + 2 \log_{3} (y) - \log_{3} (z) - \frac{3 \log_{3} (w)}{5}$