Precalculus Examples

$\ln (\frac{\sqrt{5 x - 7}}{3})$

Step 1

Rewrite $\ln (\frac{\sqrt{5 x - 7}}{3})$ as $\ln (\sqrt{5 x - 7}) - \ln (3)$ .

$\ln (\sqrt{5 x - 7}) - \ln (3)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 7}$ as ${(5 x - 7)}^{\frac{1}{2}}$ .

$\ln ({(5 x - 7)}^{\frac{1}{2}}) - \ln (3)$

Step 3

Expand $\ln ({(5 x - 7)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (5 x - 7) - \ln (3)$

Step 4

Combine $\frac{1}{2}$ and $\ln (5 x - 7)$ .

$\frac{\ln (5 x - 7)}{2} - \ln (3)$

Step 5

To write $- \ln (3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (5 x - 7)}{2} - \ln (3) \cdot \frac{2}{2}$

Step 6

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

$\frac{\ln (5 x - 7)}{2} + \frac{- \ln (3) \cdot 2}{2}$

Step 7

Combine the numerators over the common denominator.

$\frac{\ln (5 x - 7) - \ln (3) \cdot 2}{2}$

Step 8

Multiply $2$ by $- 1$ .

$\frac{\ln (5 x - 7) - 2 \ln (3)}{2}$