Trigonometry Examples

$\log (\sqrt[3]{27 y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{27 y}$ as ${(27 y)}^{\frac{1}{3}}$ .

$\log ({(27 y)}^{\frac{1}{3}})$

Step 2

Expand $\log ({(27 y)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (27 y)$

Step 3

Rewrite $\log (27 y)$ as $\log (27) + \log (y)$ .

$\frac{1}{3} (\log (27) + \log (y))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log (27)$ as $\log (3^{3})$ .

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

Step 4.2

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

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $3$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

$\log (3) + \frac{\log (y)}{3}$