Algebra Examples

$\log_{a} (32) \div \log_{a} (4)$

Step 1

Rewrite the division as a fraction.

$\frac{\log_{a} (32)}{\log_{a} (4)}$

Step 2

Rewrite $\log_{a} (32)$ 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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{a} (4)}$

Step 2.2

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

$\frac{\frac{\log (32)}{\log (a)}}{\log_{a} (4)}$

Step 3

Rewrite $\log_{a} (4)$ using the change of base formula.

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Step 3.1

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 $0$ .

$\frac{\frac{\log (32)}{\log (a)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 3.2

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

$\frac{\frac{\log (32)}{\log (a)}}{\frac{\log (4)}{\log (a)}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (32)}{\log (a)} \cdot \frac{\log (a)}{\log (4)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$\frac{\log (32)}{\log (a)} \cdot \frac{\log (a)}{\log (4)}$

Step 5.2

Rewrite the expression.

$\log (32) \frac{1}{\log (4)}$

Step 6

Combine $\log (32)$ and $\frac{1}{\log (4)}$ .

$\frac{\log (32)}{\log (4)}$

Step 7

Rewrite $\log (32)$ as $\log (2^{5})$ .

$\frac{\log (2^{5})}{\log (4)}$

Step 8

Expand $\log (2^{5})$ by moving $5$ outside the logarithm.

$\frac{5 \log (2)}{\log (4)}$

Step 9

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{5 \log (2)}{\log (2^{2})}$

Step 10

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

$\frac{5 \log (2)}{2 \log (2)}$

Step 11

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

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Step 11.1

Cancel the common factor.

$\frac{5 \log (2)}{2 \log (2)}$

Step 11.2

Rewrite the expression.

$\frac{5}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$