Precalculus Examples

$\ln (\frac{x^{4} \sqrt{y}}{z^{5}})$

Step 1

Rewrite $\ln (\frac{x^{4} \sqrt{y}}{z^{5}})$ as $\ln (x^{4} \sqrt{y}) - \ln (z^{5})$ .

$\ln (x^{4} \sqrt{y}) - \ln (z^{5})$

Step 2

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

$\ln (x^{4} y^{\frac{1}{2}}) - \ln (z^{5})$

Step 3

Expand $\ln (z^{5})$ by moving $5$ outside the logarithm.

$\ln (x^{4} y^{\frac{1}{2}}) - (5 \ln (z))$

Step 4

Multiply $5$ by $- 1$ .

$\ln (x^{4} y^{\frac{1}{2}}) - 5 \ln (z)$

Step 5

Rewrite $\ln (x^{4} y^{\frac{1}{2}})$ as $\ln (x^{4}) + \ln (y^{\frac{1}{2}})$ .

$\ln (x^{4}) + \ln (y^{\frac{1}{2}}) - 5 \ln (z)$

Step 6

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

$4 \ln (x) + \ln (y^{\frac{1}{2}}) - 5 \ln (z)$

Step 7

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

$4 \ln (x) + \frac{1}{2} \ln (y) - 5 \ln (z)$

Step 8

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

$4 \ln (x) + \frac{\ln (y)}{2} - 5 \ln (z)$