Precalculus Examples

$\log (\sqrt[3]{x^{2}} \sqrt[4]{y^{7}})$

Step 1

Rewrite $\log (\sqrt[3]{x^{2}} \sqrt[4]{y^{7}})$ as $\log (\sqrt[3]{x^{2}}) + \log (\sqrt[4]{y^{7}})$ .

$\log (\sqrt[3]{x^{2}}) + \log (\sqrt[4]{y^{7}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\log (x^{\frac{2}{3}}) + \log (\sqrt[4]{y^{7}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y^{7}}$ as $y^{\frac{7}{4}}$ .

$\log (x^{\frac{2}{3}}) + \log (y^{\frac{7}{4}})$

Step 4

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

$\frac{2}{3} \log (x) + \log (y^{\frac{7}{4}})$

Step 5

Expand $\log (y^{\frac{7}{4}})$ by moving $\frac{7}{4}$ outside the logarithm.

$\frac{2}{3} \log (x) + \frac{7}{4} \log (y)$

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{2}{3}$ and $\log (x)$ .

$\frac{2 \log (x)}{3} + \frac{7}{4} \log (y)$

Step 6.2

Combine $\frac{7}{4}$ and $\log (y)$ .

$\frac{2 \log (x)}{3} + \frac{7 \log (y)}{4}$