Algebra Examples

$\log_{12} (\frac{\frac{1}{2}}{8 w})$

Step 1

Rewrite $\log_{12} (\frac{\frac{1}{2}}{8 w})$ as $\log_{12} (\frac{1}{2}) - \log_{12} (8 w)$ .

$\log_{12} (\frac{1}{2}) - \log_{12} (8 w)$

Step 2

Rewrite $\log_{12} (\frac{1}{2})$ as $\log_{12} (1) - \log_{12} (2)$ .

$\log_{12} (1) - \log_{12} (2) - \log_{12} (8 w)$

Step 3

Logarithm base $12$ of $1$ is $0$ .

$0 - \log_{12} (2) - \log_{12} (8 w)$

Step 4

Subtract $\log_{12} (2)$ from $0$ .

$- \log_{12} (2) - \log_{12} (8 w)$

Step 5

Rewrite $\log_{12} (8 w)$ as $\log_{12} (8) + \log_{12} (w)$ .

$- \log_{12} (2) - (\log_{12} (8) + \log_{12} (w))$

Step 6

Simplify each term.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite $\log_{12} (8)$ as $\log_{12} (2^{3})$ .

$- \log_{12} (2) - (\log_{12} (2^{3}) + \log_{12} (w))$

Step 6.1.2

Expand $\log_{12} (2^{3})$ by moving $3$ outside the logarithm.

$- \log_{12} (2) - (3 \log_{12} (2) + \log_{12} (w))$

Step 6.2

Apply the distributive property.

$- \log_{12} (2) - (3 \log_{12} (2)) - \log_{12} (w)$

Step 6.3

Multiply $3$ by $- 1$ .

$- \log_{12} (2) - 3 \log_{12} (2) - \log_{12} (w)$

Step 7

Subtract $3 \log_{12} (2)$ from $- \log_{12} (2)$ .

$- 4 \log_{12} (2) - \log_{12} (w)$