Precalculus Examples

$\ln (\sqrt[4]{3}) - \ln (9)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$\ln (3^{\frac{1}{4}}) - \ln (9)$

Step 2

Expand $\ln (3^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \ln (3) - \ln (9)$

Step 3

Combine $\frac{1}{4}$ and $\ln (3)$ .

$\frac{\ln (3)}{4} - \ln (9)$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\frac{\ln (3)}{4} - \ln (3^{2})$

Step 4.2

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

$\frac{\ln (3)}{4} - (2 \ln (3))$

Step 4.3

Multiply $2$ by $- 1$ .

$\frac{\ln (3)}{4} - 2 \ln (3)$

Step 5

To write $- 2 \ln (3)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\ln (3)}{4} - 2 \ln (3) \cdot \frac{4}{4}$

Step 6

Combine $- 2 \ln (3)$ and $\frac{4}{4}$ .

$\frac{\ln (3)}{4} + \frac{- 2 \ln (3) \cdot 4}{4}$

Step 7

Combine the numerators over the common denominator.

$\frac{\ln (3) - 2 \ln (3) \cdot 4}{4}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $4$ by $- 2$ .

$\frac{\ln (3) - 8 \ln (3)}{4}$

Step 8.2

Subtract $8 \ln (3)$ from $\ln (3)$ .

$\frac{- 7 \ln (3)}{4}$

Step 9

Move the negative in front of the fraction.

$- \frac{7 \ln (3)}{4}$