Algebra Examples

$\log_{6} (x^{2} - 4) - \log_{6} (3 x + 6) = 0$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{6} (\frac{x^{2} - 4}{3 x + 6}) = 0$

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

$\log_{6} (\frac{x^{2} - 2^{2}}{3 x + 6}) = 0$

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 $b = 2$ .

$\log_{6} (\frac{(x + 2) (x - 2)}{3 x + 6}) = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

Factor $3$ out of $3 x + 6$ .

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Step 1.3.1.1

Factor $3$ out of $3 x$ .

$\log_{6} (\frac{(x + 2) (x - 2)}{3 (x) + 6}) = 0$

Step 1.3.1.2

Factor $3$ out of $6$ .

$\log_{6} (\frac{(x + 2) (x - 2)}{3 x + 3 \cdot 2}) = 0$

Step 1.3.1.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

$\log_{6} (\frac{(x + 2) (x - 2)}{3 (x + 2)}) = 0$

Step 1.3.2

Cancel the common factor of $x + 2$ .

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Step 1.3.2.1

Cancel the common factor.

$\log_{6} (\frac{(x + 2) (x - 2)}{3 (x + 2)}) = 0$

Step 1.3.2.2

Rewrite the expression.

$\log_{6} (\frac{x - 2}{3}) = 0$

Step 2

Rewrite $\log_{6} (\frac{x - 2}{3}) = 0$ 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 $b^{y} = x$ .

$6^{0} = \frac{x - 2}{3}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{x - 2}{3} = 6^{0}$ .

$\frac{x - 2}{3} = 6^{0}$

Step 3.2

Multiply both sides of the equation by $3$ .

$3 \frac{x - 2}{3} = 3 \cdot 6^{0}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

$3 \frac{x - 2}{3} = 3 \cdot 6^{0}$

Step 3.3.1.1.2

Rewrite the expression.

$x - 2 = 3 \cdot 6^{0}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $3 \cdot 6^{0}$ .

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Step 3.3.2.1.1

Anything raised to $0$ is $1$ .

$x - 2 = 3 \cdot 1$

Step 3.3.2.1.2

Multiply $3$ by $1$ .

$x - 2 = 3$

Step 3.4

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

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Step 3.4.1

Add $2$ to both sides of the equation.

$x = 3 + 2$

Step 3.4.2

Add $3$ and $2$ .

$x = 5$