Precalculus Examples

$\frac{\ln (\frac{1}{a^{4}} \cdot \frac{1}{b^{4}})}{\ln (b c)}$

Step 1

Rewrite $\ln (\frac{1}{a^{4}} \cdot \frac{1}{b^{4}})$ as $\ln (\frac{1}{a^{4}}) + \ln (\frac{1}{b^{4}})$ .

$\frac{\ln (\frac{1}{a^{4}}) + \ln (\frac{1}{b^{4}})}{\ln (b c)}$

Step 2

Rewrite $\ln (\frac{1}{a^{4}})$ as $\ln (1) - \ln (a^{4})$ .

$\frac{\ln (1) - \ln (a^{4}) + \ln (\frac{1}{b^{4}})}{\ln (b c)}$

Step 3

Rewrite $\ln (\frac{1}{b^{4}})$ as $\ln (1) - \ln (b^{4})$ .

$\frac{\ln (1) - \ln (a^{4}) + \ln (1) - \ln (b^{4})}{\ln (b c)}$

Step 4

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

$\frac{\ln (1) - (4 \ln (a)) + \ln (1) - \ln (b^{4})}{\ln (b c)}$

Step 5

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

$\frac{\ln (1) - (4 \ln (a)) + \ln (1) - (4 \ln (b))}{\ln (b c)}$

Step 6

The natural logarithm of $1$ is $0$ .

$\frac{0 - (4 \ln (a)) + \ln (1) - (4 \ln (b))}{\ln (b c)}$

Step 7

The natural logarithm of $1$ is $0$ .

$\frac{0 - (4 \ln (a)) + 0 - (4 \ln (b))}{\ln (b c)}$

Step 8

Subtract $4 \ln (a)$ from $0$ .

$\frac{- (4 \ln (a)) + 0 - (4 \ln (b))}{\ln (b c)}$

Step 9

Subtract $4 \ln (b)$ from $0$ .

$\frac{- (4 \ln (a)) - (4 \ln (b))}{\ln (b c)}$

Step 10

Multiply $4$ by $- 1$ .

$\frac{- 4 \ln (a) - (4 \ln (b))}{\ln (b c)}$

Step 11

Multiply $4$ by $- 1$ .

$\frac{- 4 \ln (a) - 4 \ln (b)}{\ln (b c)}$

Step 12

Rewrite $\ln (b c)$ as $\ln (b) + \ln (c)$ .

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

Step 13

Factor $- 4$ out of $- 4 \ln (a) - 4 \ln (b)$ .

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Step 13.1

Factor $- 4$ out of $- 4 \ln (a)$ .

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

Step 13.2

Factor $- 4$ out of $- 4 \ln (b)$ .

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

Step 13.3

Factor $- 4$ out of $- 4 (\ln (a)) - 4 (\ln (b))$ .

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

Step 14

Move the negative in front of the fraction.

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