Precalculus Examples

$\frac{\log_{2} (4 a)}{\log_{2} (2 a)}$

Step 1

Rewrite $\log_{2} (4 a)$ 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 $0$ .

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

Step 1.2

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

$\frac{\frac{\log (4 a)}{\log (2)}}{\log_{2} (2 a)}$

Step 2

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

$\frac{\frac{\log (4 a)}{\log (2)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 2.2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (4 a)}{\log (2)} \cdot \frac{\log (2)}{\log (2 a)}$

Step 4

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

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Step 4.1

Cancel the common factor.

$\frac{\log (4 a)}{\log (2)} \cdot \frac{\log (2)}{\log (2 a)}$

Step 4.2

Rewrite the expression.

$\log (4 a) \frac{1}{\log (2 a)}$

Step 5

Combine $\log (4 a)$ and $\frac{1}{\log (2 a)}$ .

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