Precalculus Examples

$\ln (\frac{x \sqrt[4]{y}}{2^{5}})$

Step 1

Rewrite $\ln (\frac{x \sqrt[4]{y}}{2^{5}})$ as $\ln (x \sqrt[4]{y}) - \ln (2^{5})$ .

$\ln (x \sqrt[4]{y}) - \ln (2^{5})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y}$ as $y^{\frac{1}{4}}$ .

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

Step 3

Expand $\ln (2^{5})$ by moving $5$ outside the logarithm.

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

Step 4

Multiply $5$ by $- 1$ .

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

Step 5

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

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

Step 6

Expand $\ln (y^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

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

Step 7

Combine $\frac{1}{4}$ and $\ln (y)$ .

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