Precalculus Examples

$\frac{\log (12)}{\log (8)} - 8$

Step 1

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{\log (8)}{\log (8)}$ .

$\frac{\log (12)}{\log (8)} - 8 \cdot \frac{\log (8)}{\log (8)}$

Step 2

Combine fractions.

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Step 2.1

Combine $- 8$ and $\frac{\log (8)}{\log (8)}$ .

$\frac{\log (12)}{\log (8)} + \frac{- 8 \log (8)}{\log (8)}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{\log (12) - 8 \log (8)}{\log (8)}$

Step 3

Simplify the numerator.

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Step 3.1

Simplify $- 8 \log (8)$ by moving $8$ inside the logarithm.

$\frac{\log (12) - \log (8^{8})}{\log (8)}$

Step 3.2

Raise $8$ to the power of $8$ .

$\frac{\log (12) - \log (16777216)}{\log (8)}$

Step 3.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\log (\frac{12}{16777216})}{\log (8)}$

Step 3.4

Cancel the common factor of $12$ and $16777216$ .

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Step 3.4.1

Factor $4$ out of $12$ .

$\frac{\log (\frac{4 (3)}{16777216})}{\log (8)}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $4$ out of $16777216$ .

$\frac{\log (\frac{4 \cdot 3}{4 \cdot 4194304})}{\log (8)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{\log (\frac{4 \cdot 3}{4 \cdot 4194304})}{\log (8)}$

Step 3.4.2.3

Rewrite the expression.

$\frac{\log (\frac{3}{4194304})}{\log (8)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{3}{4194304})}{\log (8)}$

Decimal Form:

$- 6.80501249 \dots$