Algebra Examples

$\log_{3} (b) \cdot \log_{b} (27)$

Step 1

Rewrite $\log_{3} (b)$ using the change of base formula.

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Step 1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{b} (27)$

Step 1.2

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

$\frac{\log (b)}{\log (3)} \log_{b} (27)$

Step 2

Rewrite $\log_{b} (27)$ using the change of base formula.

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Step 2.1

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 $0$ .

$\frac{\log (b)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (b)}{\log (3)} \cdot \frac{\log (27)}{\log (b)}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\frac{\log (b)}{\log (3)} \cdot \frac{\log (27)}{\log (b)}$

Step 3.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (27)$

Step 4

Combine $\frac{1}{\log (3)}$ and $\log (27)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $3$ by $1$ .

$3$