Precalculus Examples

$\log_{5} (x + 7) + \log_{5} (x - 7) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{5} ((x + 7) (x - 7)) = 3$

Step 1.2

Expand $(x + 7) (x - 7)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log_{5} (x (x - 7) + 7 (x - 7)) = 3$

Step 1.2.2

Apply the distributive property.

$\log_{5} (x \cdot x + x \cdot - 7 + 7 (x - 7)) = 3$

Step 1.2.3

Apply the distributive property.

$\log_{5} (x \cdot x + x \cdot - 7 + 7 x + 7 \cdot - 7) = 3$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $x \cdot x + x \cdot - 7 + 7 x + 7 \cdot - 7$ .

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Step 1.3.1.1

Reorder the factors in the terms $x \cdot - 7$ and $7 x$ .

$\log_{5} (x \cdot x - 7 x + 7 x + 7 \cdot - 7) = 3$

Step 1.3.1.2

Add $- 7 x$ and $7 x$ .

$\log_{5} (x \cdot x + 0 + 7 \cdot - 7) = 3$

Step 1.3.1.3

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

$\log_{5} (x \cdot x + 7 \cdot - 7) = 3$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Multiply $x$ by $x$ .

$\log_{5} (x^{2} + 7 \cdot - 7) = 3$

Step 1.3.2.2

Multiply $7$ by $- 7$ .

$\log_{5} (x^{2} - 49) = 3$

Step 2

Rewrite $\log_{5} (x^{2} - 49) = 3$ 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 $b^{y} = x$ .

$5^{3} = x^{2} - 49$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - 49 = 5^{3}$ .

$x^{2} - 49 = 5^{3}$

Step 3.2

Raise $5$ to the power of $3$ .

$x^{2} - 49 = 125$

Step 3.3

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

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Step 3.3.1

Add $49$ to both sides of the equation.

$x^{2} = 125 + 49$

Step 3.3.2

Add $125$ and $49$ .

$x^{2} = 174$

Step 3.4

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

$x = \pm \sqrt{174}$

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.

$x = \sqrt{174}$

Step 3.5.2

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

$x = - \sqrt{174}$

Step 3.5.3

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

$x = \sqrt{174}, - \sqrt{174}$

Step 4

Exclude the solutions that do not make $\log_{5} (x + 7) + \log_{5} (x - 7) = 3$ true.

$x = \sqrt{174}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{174}$

Decimal Form:

$x = 13.19090595 \dots$