Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $4$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Move the negative in front of the fraction.

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