Algebra Examples

$\log_{a} (x) \div \log_{a b} (x)$

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 2.2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (x)}{\log (a)} \cdot \frac{1}{\log_{a b} (x)}$

Step 4

Multiply $\frac{\log (x)}{\log (a)}$ by $\frac{1}{\log_{a b} (x)}$ .

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