Precalculus Examples

$\log_{6} (\frac{1}{6}) + \log_{6} (\sqrt{6}) + \log_{6} (36)$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{6} (\frac{1}{6} \sqrt{6}) + \log_{6} (36)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{6} (\frac{1}{6} \sqrt{6} \cdot 36)$

Step 3

Combine $\frac{1}{6}$ and $\sqrt{6}$ .

$\log_{6} (\frac{\sqrt{6}}{6} \cdot 36)$

Step 4

Cancel the common factor of $6$ .

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Step 4.1

Factor $6$ out of $36$ .

$\log_{6} (\frac{\sqrt{6}}{6} \cdot (6 (6)))$

Step 4.2

Cancel the common factor.

$\log_{6} (\frac{\sqrt{6}}{6} \cdot (6 \cdot 6))$

Step 4.3

Rewrite the expression.

$\log_{6} (\sqrt{6} \cdot 6)$

Step 5

Move $6$ to the left of $\sqrt{6}$ .

$\log_{6} (6 \sqrt{6})$

Step 6

Rewrite $\log_{6} (6 \sqrt{6})$ using the change of base formula.

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Step 6.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 6.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (6 \sqrt{6})}{\log (6)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (6 \sqrt{6})}{\log (6)}$

Decimal Form:

$1.5$