Precalculus Examples

$\log_{a} (\sqrt[3]{x y^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x y^{3}}$ as ${(x y^{3})}^{\frac{1}{3}}$ .

$\log_{a} ({(x y^{3})}^{\frac{1}{3}})$

Step 2

Expand $\log_{a} ({(x y^{3})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{a} (x y^{3})$

Step 3

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

$\frac{1}{3} (\log_{a} (x) + \log_{a} (y^{3}))$

Step 4

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

$\frac{1}{3} (\log_{a} (x) + 3 \log_{a} (y))$

Step 5

Apply the distributive property.

$\frac{1}{3} \log_{a} (x) + \frac{1}{3} (3 \log_{a} (y))$

Step 6

Combine $\frac{1}{3}$ and $\log_{a} (x)$ .

$\frac{\log_{a} (x)}{3} + \frac{1}{3} (3 \log_{a} (y))$

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Factor $3$ out of $3 \log_{a} (y)$ .

$\frac{\log_{a} (x)}{3} + \frac{1}{3} (3 (\log_{a} (y)))$

Step 7.2

Cancel the common factor.

$\frac{\log_{a} (x)}{3} + \frac{1}{3} (3 \log_{a} (y))$

Step 7.3

Rewrite the expression.

$\frac{\log_{a} (x)}{3} + \log_{a} (y)$