Algebra Examples

$\log (x + 2) + \log (x - 2) = 1 - \log (2)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((x + 2) (x - 2)) = 1 - \log (2)$

Step 1.2

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log (x (x - 2) + 2 (x - 2)) = 1 - \log (2)$

Step 1.2.2

Apply the distributive property.

$\log (x \cdot x + x \cdot - 2 + 2 (x - 2)) = 1 - \log (2)$

Step 1.2.3

Apply the distributive property.

$\log (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2) = 1 - \log (2)$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 1.3.1.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$\log (x \cdot x - 2 x + 2 x + 2 \cdot - 2) = 1 - \log (2)$

Step 1.3.1.2

Add $- 2 x$ and $2 x$ .

$\log (x \cdot x + 0 + 2 \cdot - 2) = 1 - \log (2)$

Step 1.3.1.3

Add $x \cdot x$ and $0$ .

$\log (x \cdot x + 2 \cdot - 2) = 1 - \log (2)$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Multiply $x$ by $x$ .

$\log (x^{2} + 2 \cdot - 2) = 1 - \log (2)$

Step 1.3.2.2

Multiply $2$ by $- 2$ .

$\log (x^{2} - 4) = 1 - \log (2)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\log (x^{2} - 4) + \log (2) = 1$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((x^{2} - 4) \cdot 2) = 1$

Step 4

Apply the distributive property.

$\log (x^{2} \cdot 2 - 4 \cdot 2) = 1$

Step 5

Simplify the expression.

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Step 5.1

Move $2$ to the left of $x^{2}$ .

$\log (2 \cdot x^{2} - 4 \cdot 2) = 1$

Step 5.2

Multiply $- 4$ by $2$ .

$\log (2 x^{2} - 8) = 1$

Step 6

Rewrite $\log (2 x^{2} - 8) = 1$ 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 $b^{y} = x$ .

$10^{1} = 2 x^{2} - 8$

Step 7

Solve for $x$ .

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Step 7.1

Rewrite the equation as $2 x^{2} - 8 = 10^{1}$ .

$2 x^{2} - 8 = 10$

Step 7.2

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

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Step 7.2.1

Add $8$ to both sides of the equation.

$2 x^{2} = 10 + 8$

Step 7.2.2

Add $10$ and $8$ .

$2 x^{2} = 18$

Step 7.3

Divide each term in $2 x^{2} = 18$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x^{2} = 18$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{18}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{18}{2}$

Step 7.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{18}{2}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $18$ by $2$ .

$x^{2} = 9$

Step 7.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 7.5

Simplify $\pm \sqrt{9}$ .

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Step 7.5.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 7.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 7.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 7.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 7.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 8

Exclude the solutions that do not make $\log (x + 2) + \log (x - 2) = 1 - \log (2)$ true.

$x = 3$