Calculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 1.3.1.1

Reorder the factors in the terms $x \cdot - 4$ and $4 x$ .

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

Step 1.3.1.2

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

$\log_{2} (x \cdot x + 0 + 4 \cdot - 4) = 1$

Step 1.3.1.3

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

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

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Multiply $x$ by $x$ .

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

Step 1.3.2.2

Multiply $4$ by $- 4$ .

$\log_{2} (x^{2} - 16) = 1$

Step 2

Rewrite $\log_{2} (x^{2} - 16) = 1$ 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 $b^{y} = x$ .

$2^{1} = x^{2} - 16$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - 16 = 2^{1}$ .

$x^{2} - 16 = 2$

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 $16$ to both sides of the equation.

$x^{2} = 2 + 16$

Step 3.2.2

Add $2$ and $16$ .

$x^{2} = 18$

Step 3.3

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

$x = \pm \sqrt{18}$

Step 3.4

Simplify $\pm \sqrt{18}$ .

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Step 3.4.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.4.1.1

Factor $9$ out of $18$ .

$x = \pm \sqrt{9 (2)}$

Step 3.4.1.2

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

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

Step 3.4.2

Pull terms out from under the radical.

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

Step 3.5

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

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Step 3.5.1

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

$x = 3 \sqrt{2}$

Step 3.5.2

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

$x = - 3 \sqrt{2}$

Step 3.5.3

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

$x = 3 \sqrt{2}, - 3 \sqrt{2}$

Step 4

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

$x = 3 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{2}$

Decimal Form:

$x = 4.24264068 \dots$