Algebra Examples

$\frac{\log_{b} (8)}{\log_{b} (2)}$

Step 1

Rewrite $\log_{b} (8)$ 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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{b} (2)}$

Step 1.2

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

$\frac{\frac{\log (8)}{\log (b)}}{\log_{b} (2)}$

Step 2

Rewrite $\log_{b} (2)$ 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 $0$ .

$\frac{\frac{\log (8)}{\log (b)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 2.2

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

$\frac{\frac{\log (8)}{\log (b)}}{\frac{\log (2)}{\log (b)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (8)}{\log (b)} \cdot \frac{\log (b)}{\log (2)}$

Step 4

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

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Step 4.1

Cancel the common factor.

$\frac{\log (8)}{\log (b)} \cdot \frac{\log (b)}{\log (2)}$

Step 4.2

Rewrite the expression.

$\log (8) \frac{1}{\log (2)}$

Step 5

Combine $\log (8)$ and $\frac{1}{\log (2)}$ .

$\frac{\log (8)}{\log (2)}$

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $3$ by $1$ .

$3$