Precalculus Examples

$\log (\sqrt[3]{\frac{8 x}{y}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{8 x}{y}}$ as ${(\frac{8 x}{y})}^{\frac{1}{3}}$ .

$\log ({(\frac{8 x}{y})}^{\frac{1}{3}})$

Step 2

Expand $\log ({(\frac{8 x}{y})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (\frac{8 x}{y})$

Step 3

Rewrite $\log (\frac{8 x}{y})$ as $\log (8 x) - \log (y)$ .

$\frac{1}{3} (\log (8 x) - \log (y))$

Step 4

Rewrite $\log (8 x)$ as $\log (8) + \log (x)$ .

$\frac{1}{3} (\log (8) + \log (x) - \log (y))$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{1}{3} (\log (2^{3}) + \log (x) - \log (y))$

Step 5.2

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

$\frac{1}{3} (3 \log (2) + \log (x) - \log (y))$

Step 6

Apply the distributive property.

$\frac{1}{3} (3 \log (2)) + \frac{1}{3} \log (x) + \frac{1}{3} (- \log (y))$

Step 7

Simplify.

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Step 7.1

Cancel the common factor of $3$ .

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Step 7.1.1

Factor $3$ out of $3 \log (2)$ .

$\frac{1}{3} (3 (\log (2))) + \frac{1}{3} \log (x) + \frac{1}{3} (- \log (y))$

Step 7.1.2

Cancel the common factor.

$\frac{1}{3} (3 \log (2)) + \frac{1}{3} \log (x) + \frac{1}{3} (- \log (y))$

Step 7.1.3

Rewrite the expression.

$\log (2) + \frac{1}{3} \log (x) + \frac{1}{3} (- \log (y))$

Step 7.2

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

$\log (2) + \frac{\log (x)}{3} + \frac{1}{3} (- \log (y))$

Step 7.3

Combine $\frac{1}{3}$ and $\log (y)$ .

$\log (2) + \frac{\log (x)}{3} - \frac{\log (y)}{3}$