Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Expand $(2 x + 4) (x - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

$\log (2 x \cdot x + 2 x \cdot - 2 + 4 (x - 2)) = 1$

Step 1.2.3

Apply the distributive property.

$\log (2 x \cdot x + 2 x \cdot - 2 + 4 x + 4 \cdot - 2) = 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.1.1

Move $x$ .

$\log (2 (x \cdot x) + 2 x \cdot - 2 + 4 x + 4 \cdot - 2) = 1$

Step 1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Multiply $- 2$ by $2$ .

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

Step 1.3.1.3

Multiply $4$ by $- 2$ .

$\log (2 x^{2} - 4 x + 4 x - 8) = 1$

Step 1.3.2

Add $- 4 x$ and $4 x$ .

$\log (2 x^{2} + 0 - 8) = 1$

Step 1.3.3

Add $2 x^{2}$ and $0$ .

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

Step 2

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 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $8$ to both sides of the equation.

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

Step 3.2.2

Add $10$ and $8$ .

$2 x^{2} = 18$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

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

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $18$ by $2$ .

$x^{2} = 9$

Step 3.4

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

$x = \pm \sqrt{9}$

Step 3.5

Simplify $\pm \sqrt{9}$ .

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Step 3.5.1

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

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

Step 3.5.2

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

$x = \pm 3$

Step 3.6

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

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Step 3.6.1

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

$x = 3$

Step 3.6.2

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

$x = - 3$

Step 3.6.3

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

$x = 3, - 3$

Step 4

Exclude the solutions that do not make $\log (2 x + 4) + \log (x - 2) = 1$ true.

$x = 3$