Precalculus Examples

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

Step 1

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

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

Step 2

Expand $\ln (a^{- 3})$ by moving $- 3$ outside the logarithm.

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

Step 3

Expand $\ln (b^{- 3})$ by moving $- 3$ outside the logarithm.

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

Step 4

Expand $\ln ({(b c)}^{3})$ by moving $3$ outside the logarithm.

$\frac{- 3 \ln (a) - 3 \ln (b)}{3 \ln (b c)}$

Step 5

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

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

Step 6

Cancel the common factor of $- 3 \ln (a) - 3 \ln (b)$ and $3$ .

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Step 6.1

Factor $3$ out of $- 3 \ln (a)$ .

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

Step 6.2

Factor $3$ out of $- 3 \ln (b)$ .

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

Step 6.3

Factor $3$ out of $3 (- \ln (a)) + 3 (- \ln (b))$ .

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

Step 6.4

Cancel the common factors.

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 7

Factor $- 1$ out of $- \ln (a) - \ln (b)$ .

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Step 7.1

Factor $- 1$ out of $- \ln (a)$ .

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

Step 7.2

Factor $- 1$ out of $- \ln (b)$ .

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

Step 7.3

Factor $- 1$ out of $- (\ln (a)) - (\ln (b))$ .

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

Step 8

Move the negative in front of the fraction.

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