Precalculus Examples

$\ln (\frac{3 a^{4} b^{7} c}{\sqrt[4]{b - 9}})$

Step 1

Rewrite $\ln (\frac{3 a^{4} b^{7} c}{\sqrt[4]{b - 9}})$ as $\ln (3 a^{4} b^{7} c) - \ln (\sqrt[4]{b - 9})$ .

$\ln (3 a^{4} b^{7} c) - \ln (\sqrt[4]{b - 9})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{b - 9}$ as ${(b - 9)}^{\frac{1}{4}}$ .

$\ln (3 a^{4} b^{7} c) - \ln ({(b - 9)}^{\frac{1}{4}})$

Step 3

Expand $\ln ({(b - 9)}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\ln (3 a^{4} b^{7} c) - (\frac{1}{4} \ln (b - 9))$

Step 4

Combine $\frac{1}{4}$ and $\ln (b - 9)$ .

$\ln (3 a^{4} b^{7} c) - \frac{\ln (b - 9)}{4}$

Step 5

Rewrite $\ln (3 a^{4} b^{7} c)$ as $\ln (3 a^{4} b^{7}) + \ln (c)$ .

$\ln (3 a^{4} b^{7}) + \ln (c) - \frac{\ln (b - 9)}{4}$

Step 6

Rewrite $\ln (3 a^{4} b^{7})$ as $\ln (3 a^{4}) + \ln (b^{7})$ .

$\ln (3 a^{4}) + \ln (b^{7}) + \ln (c) - \frac{\ln (b - 9)}{4}$

Step 7

Rewrite $\ln (3 a^{4})$ as $\ln (3) + \ln (a^{4})$ .

$\ln (3) + \ln (a^{4}) + \ln (b^{7}) + \ln (c) - \frac{\ln (b - 9)}{4}$

Step 8

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

$\ln (3) + 4 \ln (a) + \ln (b^{7}) + \ln (c) - \frac{\ln (b - 9)}{4}$

Step 9

Expand $\ln (b^{7})$ by moving $7$ outside the logarithm.

$\ln (3) + 4 \ln (a) + 7 \ln (b) + \ln (c) - \frac{\ln (b - 9)}{4}$

Step 10

To write $\ln (3)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4 \ln (a) + 7 \ln (b) + \ln (c) + \ln (3) \cdot \frac{4}{4} - \frac{\ln (b - 9)}{4}$

Step 11

Combine $\ln (3)$ and $\frac{4}{4}$ .

$4 \ln (a) + 7 \ln (b) + \ln (c) + \frac{\ln (3) \cdot 4}{4} - \frac{\ln (b - 9)}{4}$

Step 12

Combine the numerators over the common denominator.

$4 \ln (a) + 7 \ln (b) + \ln (c) + \frac{\ln (3) \cdot 4 - \ln (b - 9)}{4}$

Step 13

Move $4$ to the left of $\ln (3)$ .

$4 \ln (a) + 7 \ln (b) + \ln (c) + \frac{4 \ln (3) - \ln (b - 9)}{4}$