Precalculus Examples

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

Step 1

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

$\ln (x y^{4}) - \ln (\sqrt{y - 1})$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

To write $\ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Move $2$ to the left of $\ln (x)$ .

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