Precalculus Examples

$\ln (\frac{1}{\sqrt{e^{3}}})$

Step 1

Rewrite $\ln (\frac{1}{\sqrt{e^{3}}})$ as $\ln (1) - \ln (\sqrt{e^{3}})$ .

$\ln (1) - \ln (\sqrt{e^{3}})$

Step 2

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

$\ln (1) - \ln (e^{\frac{3}{2}})$

Step 3

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

$\ln (1) - (\frac{3}{2} \ln (e))$

Step 4

The natural logarithm of $1$ is $0$ .

$0 - (\frac{3}{2} \ln (e))$

Step 5

The natural logarithm of $e$ is $1$ .

$0 - (\frac{3}{2} \cdot 1)$

Step 6

Multiply $\frac{3}{2}$ by $1$ .

$0 - \frac{3}{2}$

Step 7

Subtract $\frac{3}{2}$ from $0$ .

$- \frac{3}{2}$