Precalculus Examples

$\ln (\frac{e^{4} x^{2}}{\sqrt{y}})$

Step 1

Rewrite $\ln (\frac{e^{4} x^{2}}{\sqrt{y}})$ as $\ln (e^{4} x^{2}) - \ln (\sqrt{y})$ .

$\ln (e^{4} x^{2}) - \ln (\sqrt{y})$

Step 2

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

$\ln (e^{4} x^{2}) - \ln (y^{\frac{1}{2}})$

Step 3

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

$\ln (e^{4} x^{2}) - (\frac{1}{2} \ln (y))$

Step 4

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

$\ln (e^{4} x^{2}) - \frac{\ln (y)}{2}$

Step 5

Rewrite $\ln (e^{4} x^{2})$ as $\ln (e^{4}) + \ln (x^{2})$ .

$\ln (e^{4}) + \ln (x^{2}) - \frac{\ln (y)}{2}$

Step 6

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

$4 \ln (e) + \ln (x^{2}) - \frac{\ln (y)}{2}$

Step 7

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

$4 \ln (e) + 2 \ln (x) - \frac{\ln (y)}{2}$

Step 8

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

$4 \cdot 1 + 2 \ln (x) - \frac{\ln (y)}{2}$

Step 9

Multiply $4$ by $1$ .

$4 + 2 \ln (x) - \frac{\ln (y)}{2}$