Precalculus Examples

$\log_{4} (x^{2} - 4) - \log_{4} (x - 2)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{x^{2} - 4}{x - 2})$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $4$ as $2^{2}$ .

$\log_{4} (\frac{x^{2} - 2^{2}}{x - 2})$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\log_{4} (\frac{(x + 2) (x - 2)}{x - 2})$

Step 3

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 3.1

Cancel the common factor.

$\log_{4} (\frac{(x + 2) (x - 2)}{x - 2})$

Step 3.2

Divide $x + 2$ by $1$ .

$\log_{4} (x + 2)$

Step 4

Rewrite $\log_{4} (x + 2)$ using the change of base formula.

Tap for more steps...

Step 4.1

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 $0$ .

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

Step 4.2

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

$\frac{\log (x + 2)}{\log (4)}$