Precalculus Examples

$\frac{\log (18) - \log (2)}{\log (27)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\log (\frac{18}{2})}{\log (27)}$

Step 1.2

Divide $18$ by $2$ .

$\frac{\log (9)}{\log (27)}$

Step 2

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\frac{\log (3^{2})}{\log (27)}$

Step 3

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

$\frac{2 \log (3)}{\log (27)}$

Step 4

Rewrite $\log (27)$ as $\log (3^{3})$ .

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

Step 5

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

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

Step 6

Cancel the common factor of $\log (3)$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$