Precalculus Examples

$\log_{2} (x^{2} - 36) - \log_{2} (x + 6) = 1$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{2} (\frac{x^{2} - 36}{x + 6}) = 1$

Step 1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.1

Rewrite $36$ as $6^{2}$ .

$\log_{2} (\frac{x^{2} - 6^{2}}{x + 6}) = 1$

Step 1.2.2

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 $b = 6$ .

$\log_{2} (\frac{(x + 6) (x - 6)}{x + 6}) = 1$

Step 1.3

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\log_{2} (\frac{(x + 6) (x - 6)}{x + 6}) = 1$

Step 1.3.2

Divide $x - 6$ by $1$ .

$\log_{2} (x - 6) = 1$

Step 2

Rewrite $\log_{2} (x - 6) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{1} = x - 6$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $x - 6 = 2^{1}$ .

$x - 6 = 2$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 3.2.1

Add $6$ to both sides of the equation.

$x = 2 + 6$

Step 3.2.2

Add $2$ and $6$ .

$x = 8$