Precalculus Examples

$\ln (\frac{\sqrt{2} x^{\frac{3}{2}}}{e^{2}})$

Step 1

Rewrite $\ln (\frac{\sqrt{2} x^{\frac{3}{2}}}{e^{2}})$ as $\ln (\sqrt{2} x^{\frac{3}{2}}) - \ln (e^{2})$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $1$ .

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

Step 6

Multiply $- 1$ by $2$ .

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

Step 7

Rewrite $\ln (2^{\frac{1}{2}} x^{\frac{3}{2}})$ as $\ln (2^{\frac{1}{2}}) + \ln (x^{\frac{3}{2}})$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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