Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Multiply $y^{\frac{1}{2}}$ by $1$ .

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

Step 11

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

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