Algebra Examples

$\log_{2} (x^{2} - 4) - \log_{2} (2 x + 4) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{2} (\frac{x^{2} - 4}{2 x + 4}) = 3$

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

$\log_{2} (\frac{x^{2} - 2^{2}}{2 x + 4}) = 3$

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 $b = 2$ .

$\log_{2} (\frac{(x + 2) (x - 2)}{2 x + 4}) = 3$

Step 1.3

Simplify terms.

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Step 1.3.1

Factor $2$ out of $2 x + 4$ .

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Step 1.3.1.1

Factor $2$ out of $2 x$ .

$\log_{2} (\frac{(x + 2) (x - 2)}{2 (x) + 4}) = 3$

Step 1.3.1.2

Factor $2$ out of $4$ .

$\log_{2} (\frac{(x + 2) (x - 2)}{2 x + 2 \cdot 2}) = 3$

Step 1.3.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\log_{2} (\frac{(x + 2) (x - 2)}{2 (x + 2)}) = 3$

Step 1.3.2

Reduce the expression $\frac{(x + 2) (x - 2)}{2 (x + 2)}$ by cancelling the common factors.

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Step 1.3.2.1

Cancel the common factor.

$\log_{2} (\frac{(x + 2) (x - 2)}{2 (x + 2)}) = 3$

Step 1.3.2.2

Rewrite the expression.

$\log_{2} (\frac{x - 2}{2}) = 3$

Step 1.4

Rewrite $\frac{x - 2}{2}$ as $(x - 2) \cdot 2^{- 1}$ .

$\log_{2} ((x - 2) \cdot 2^{- 1}) = 3$

Step 1.5

Rewrite $\log_{2} ((x - 2) \cdot 2^{- 1})$ as $\log_{2} (x - 2) + \log_{2} (2^{- 1})$ .

$\log_{2} (x - 2) + \log_{2} (2^{- 1}) = 3$

Step 1.6

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{2} (x - 2) - \log_{2} (2) = 3$

Step 1.7

Logarithm base $2$ of $2$ is $1$ .

$\log_{2} (x - 2) - 1 \cdot 1 = 3$

Step 1.8

Multiply $- 1$ by $1$ .

$\log_{2} (x - 2) - 1 = 3$

Step 2

Move all terms not containing $\log_{2} (x - 2)$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$\log_{2} (x - 2) = 3 + 1$

Step 2.2

Add $3$ and $1$ .

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

Step 3

Rewrite $\log_{2} (x - 2) = 4$ 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 $b^{y} = x$ .

$2^{4} = x - 2$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x - 2 = 2^{4}$ .

$x - 2 = 2^{4}$

Step 4.2

Raise $2$ to the power of $4$ .

$x - 2 = 16$

Step 4.3

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

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Step 4.3.1

Add $2$ to both sides of the equation.

$x = 16 + 2$

Step 4.3.2

Add $16$ and $2$ .

$x = 18$