Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Multiply $5$ by $- 1$ .

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

Step 8

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

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