Precalculus Examples

$\ln (\frac{\sqrt{3 x - 5}}{y^{2}})$

Step 1

Rewrite $\ln (\frac{\sqrt{3 x - 5}}{y^{2}})$ as $\ln (\sqrt{3 x - 5}) - \ln (y^{2})$ .

$\ln (\sqrt{3 x - 5}) - \ln (y^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 5}$ as ${(3 x - 5)}^{\frac{1}{2}}$ .

$\ln ({(3 x - 5)}^{\frac{1}{2}}) - \ln (y^{2})$

Step 3

Expand $\ln ({(3 x - 5)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (3 x - 5) - \ln (y^{2})$

Step 4

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

$\frac{1}{2} \ln (3 x - 5) - (2 \ln (y))$

Step 5

Multiply $2$ by $- 1$ .

$\frac{1}{2} \ln (3 x - 5) - 2 \ln (y)$

Step 6

Combine $\frac{1}{2}$ and $\ln (3 x - 5)$ .

$\frac{\ln (3 x - 5)}{2} - 2 \ln (y)$

Step 7

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

$\frac{\ln (3 x - 5)}{2} - 2 \ln (y) \cdot \frac{2}{2}$

Step 8

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

$\frac{\ln (3 x - 5)}{2} + \frac{- 2 \ln (y) \cdot 2}{2}$

Step 9

Combine the numerators over the common denominator.

$\frac{\ln (3 x - 5) - 2 \ln (y) \cdot 2}{2}$

Step 10

Multiply $2$ by $- 2$ .

$\frac{\ln (3 x - 5) - 4 \ln (y)}{2}$