Precalculus Examples

$\ln (\frac{5 \sqrt{x}}{6})$

Step 1

Rewrite $\ln (\frac{5 \sqrt{x}}{6})$ as $\ln (5 \sqrt{x}) - \ln (6)$ .

$\ln (5 \sqrt{x}) - \ln (6)$

Step 2

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

$\ln (5 x^{\frac{1}{2}}) - \ln (6)$

Step 3

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

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

Step 4

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

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

Step 5

Apply the distributive property.

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

Step 6

Rewrite $\ln (5 x^{\frac{1}{2}})$ as $\ln (5) + \ln (x^{\frac{1}{2}})$ .

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

Step 7

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

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

Step 8

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

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