Precalculus Examples

$\ln (x \sqrt[9]{\frac{y}{z}})$

Step 1

Rewrite $\ln (x \sqrt[9]{\frac{y}{z}})$ as $\ln (x) + \ln (\sqrt[9]{\frac{y}{z}})$ .

$\ln (x) + \ln (\sqrt[9]{\frac{y}{z}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{\frac{y}{z}}$ as ${(\frac{y}{z})}^{\frac{1}{9}}$ .

$\ln (x) + \ln ({(\frac{y}{z})}^{\frac{1}{9}})$

Step 3

Expand $\ln ({(\frac{y}{z})}^{\frac{1}{9}})$ by moving $\frac{1}{9}$ outside the logarithm.

$\ln (x) + \frac{1}{9} \ln (\frac{y}{z})$

Step 4

Rewrite $\ln (\frac{y}{z})$ as $\ln (y) - \ln (z)$ .

$\ln (x) + \frac{1}{9} (\ln (y) - \ln (z))$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\ln (x) + \frac{1}{9} \ln (y) + \frac{1}{9} (- \ln (z))$

Step 5.2

Combine $\frac{1}{9}$ and $\ln (y)$ .

$\ln (x) + \frac{\ln (y)}{9} + \frac{1}{9} (- \ln (z))$

Step 5.3

Combine $\frac{1}{9}$ and $\ln (z)$ .

$\ln (x) + \frac{\ln (y)}{9} - \frac{\ln (z)}{9}$