Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $- 3$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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