Precalculus Examples

$\log (\frac{\sqrt{x}}{\sqrt[3]{y^{5}}})$

Step 1

Rewrite $\log (\frac{\sqrt{x}}{\sqrt[3]{y^{5}}})$ as $\log (\sqrt{x}) - \log (\sqrt[3]{y^{5}})$ .

$\log (\sqrt{x}) - \log (\sqrt[3]{y^{5}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log (x^{\frac{1}{2}}) - \log (\sqrt[3]{y^{5}})$

Step 3

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

$\log (x^{\frac{1}{2}}) - \log (y^{\frac{5}{3}})$

Step 4

Expand $\log (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (x) - \log (y^{\frac{5}{3}})$

Step 5

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

$\frac{1}{2} \log (x) - (\frac{5}{3} \log (y))$

Step 6

Combine $\frac{5}{3}$ and $\log (y)$ .

$\frac{1}{2} \log (x) - \frac{5 \log (y)}{3}$

Step 7

Combine $\frac{1}{2}$ and $\log (x)$ .

$\frac{\log (x)}{2} - \frac{5 \log (y)}{3}$