Precalculus Examples

$\frac{\log (\frac{1}{8})}{\log (\sqrt{2})}$

Step 1

Rewrite $\log (\frac{1}{8})$ as $\log (1) - \log (8)$ .

$\frac{\log (1) - \log (8)}{\log (\sqrt{2})}$

Step 2

Logarithm base $10$ of $1$ is $0$ .

$\frac{0 - \log (8)}{\log (\sqrt{2})}$

Step 3

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{0 - \log (2^{3})}{\log (\sqrt{2})}$

Step 4

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$\frac{0 - (3 \log (2))}{\log (\sqrt{2})}$

Step 5

Multiply $3$ by $- 1$ .

$\frac{0 - 3 \log (2)}{\log (\sqrt{2})}$

Step 6

Subtract $3 \log (2)$ from $0$ .

$\frac{- 3 \log (2)}{\log (\sqrt{2})}$

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{- 3 \log (2)}{\log (2^{\frac{1}{2}})}$

Step 8

Expand $\log (2^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{- 3 \log (2)}{\frac{1}{2} \log (2)}$

Step 9

Combine $\frac{1}{2}$ and $\log (2)$ .

$\frac{- 3 \log (2)}{\frac{\log (2)}{2}}$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$- 3 \log (2) \frac{2}{\log (2)}$

Step 11

Cancel the common factor of $\log (2)$ .

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Step 11.1

Factor $\log (2)$ out of $- 3 \log (2)$ .

$\log (2) \cdot - 3 \frac{2}{\log (2)}$

Step 11.2

Cancel the common factor.

$\log (2) \cdot - 3 \frac{2}{\log (2)}$

Step 11.3

Rewrite the expression.

$- 3 \cdot 2$

Step 12

Multiply $- 3$ by $2$ .

$- 6$