Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

Factor $3$ out of $3 x^{2} - 3$ .

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Step 1.2.1.1

Factor $3$ out of $3 x^{2}$ .

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

Step 1.2.1.2

Factor $3$ out of $- 3$ .

$\log_{6} (\frac{3 (x^{2}) + 3 (- 1)}{4 x + 4}) = 0$

Step 1.2.1.3

Factor $3$ out of $3 (x^{2}) + 3 (- 1)$ .

$\log_{6} (\frac{3 (x^{2} - 1)}{4 x + 4}) = 0$

Step 1.2.2

Rewrite $1$ as $1^{2}$ .

$\log_{6} (\frac{3 (x^{2} - 1^{2})}{4 x + 4}) = 0$

Step 1.2.3

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 $b = 1$ .

$\log_{6} (\frac{3 (x + 1) (x - 1)}{4 x + 4}) = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

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

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Step 1.3.1.1

Factor $4$ out of $4 x$ .

$\log_{6} (\frac{3 (x + 1) (x - 1)}{4 (x) + 4}) = 0$

Step 1.3.1.2

Factor $4$ out of $4$ .

$\log_{6} (\frac{3 (x + 1) (x - 1)}{4 (x) + 4 (1)}) = 0$

Step 1.3.1.3

Factor $4$ out of $4 (x) + 4 (1)$ .

$\log_{6} (\frac{3 (x + 1) (x - 1)}{4 (x + 1)}) = 0$

Step 1.3.2

Cancel the common factor of $x + 1$ .

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Step 1.3.2.1

Cancel the common factor.

$\log_{6} (\frac{3 (x + 1) (x - 1)}{4 (x + 1)}) = 0$

Step 1.3.2.2

Rewrite the expression.

$\log_{6} (\frac{3 (x - 1)}{4}) = 0$

Step 2

Rewrite $\log_{6} (\frac{3 (x - 1)}{4}) = 0$ 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 $b^{y} = x$ .

$6^{0} = \frac{3 (x - 1)}{4}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{3 (x - 1)}{4} = 6^{0}$ .

$\frac{3 (x - 1)}{4} = 6^{0}$

Step 3.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 (x - 1)}{4} = \frac{4}{3} \cdot 6^{0}$

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

Simplify $\frac{4}{3} \cdot \frac{3 (x - 1)}{4}$ .

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 (x - 1)}{4} = \frac{4}{3} \cdot 6^{0}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 (x - 1)) = \frac{4}{3} \cdot 6^{0}$

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{1}{3} (3 (x - 1)) = \frac{4}{3} \cdot 6^{0}$

Step 3.3.1.1.2.2

Rewrite the expression.

$x - 1 = \frac{4}{3} \cdot 6^{0}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{4}{3} \cdot 6^{0}$ .

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Step 3.3.2.1.1

Anything raised to $0$ is $1$ .

$x - 1 = \frac{4}{3} \cdot 1$

Step 3.3.2.1.2

Multiply $\frac{4}{3}$ by $1$ .

$x - 1 = \frac{4}{3}$

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 $1$ to both sides of the equation.

$x = \frac{4}{3} + 1$

Step 3.4.2

Write $1$ as a fraction with a common denominator.

$x = \frac{4}{3} + \frac{3}{3}$

Step 3.4.3

Combine the numerators over the common denominator.

$x = \frac{4 + 3}{3}$

Step 3.4.4

Add $4$ and $3$ .

$x = \frac{7}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{3}$

Decimal Form:

$x = 2.\overline{3}$

Mixed Number Form:

$x = 2 \frac{1}{3}$