Precalculus Examples

$(\log (4) - \log (8)) (\log (10) \div \log (2))$

Step 1

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

$\log (\frac{4}{8}) \log (10) \div \log (2)$

Step 2

Cancel the common factor of $4$ and $8$ .

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Step 2.1

Factor $4$ out of $4$ .

$\log (\frac{4 (1)}{8}) \log (10) \div \log (2)$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $4$ out of $8$ .

$\log (\frac{4 \cdot 1}{4 \cdot 2}) \log (10) \div \log (2)$

Step 2.2.2

Cancel the common factor.

$\log (\frac{4 \cdot 1}{4 \cdot 2}) \log (10) \div \log (2)$

Step 2.2.3

Rewrite the expression.

$\log (\frac{1}{2}) \log (10) \div \log (2)$

Step 3

Rewrite the division as a fraction.

$\log (\frac{1}{2}) \frac{\log (10)}{\log (2)}$

Step 4

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

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

Step 5

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

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

Step 6

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $- 1$ by $1$ .

$- 1$