Algebra Examples

$\log_{5} (x^{2} - 4) - \log_{5} (x + 4) = 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{5} (\frac{x^{2} - 4}{x + 4}) = 1$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $4$ as $2^{2}$ .

$\log_{5} (\frac{x^{2} - 2^{2}}{x + 4}) = 1$

Step 1.2.2

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 $b = 2$ .

$\log_{5} (\frac{(x + 2) (x - 2)}{x + 4}) = 1$

Step 2

Rewrite $\log_{5} (\frac{(x + 2) (x - 2)}{x + 4}) = 1$ 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 $b^{y} = x$ .

$5^{1} = \frac{(x + 2) (x - 2)}{x + 4}$

Step 3

Cross multiply to remove the fraction.

$(x + 2) (x - 2) = 5 (x + 4)$

Step 4

Simplify $5 (x + 4)$ .

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Step 4.1

Apply the distributive property.

$(x + 2) (x - 2) = 5 x + 5 \cdot 4$

Step 4.2

Multiply $5$ by $4$ .

$(x + 2) (x - 2) = 5 x + 20$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $5 x$ from both sides of the equation.

$(x + 2) (x - 2) - 5 x = 20$

Step 5.2

Simplify each term.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the distributive property.

$x (x - 2) + 2 (x - 2) - 5 x = 20$

Step 5.2.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 2 (x - 2) - 5 x = 20$

Step 5.2.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2 - 5 x = 20$

Step 5.2.2

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

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Step 5.2.2.1

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

$x \cdot x - 2 x + 2 x + 2 \cdot - 2 - 5 x = 20$

Step 5.2.2.2

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

$x \cdot x + 0 + 2 \cdot - 2 - 5 x = 20$

Step 5.2.2.3

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

$x \cdot x + 2 \cdot - 2 - 5 x = 20$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply $x$ by $x$ .

$x^{2} + 2 \cdot - 2 - 5 x = 20$

Step 5.2.3.2

Multiply $2$ by $- 2$ .

$x^{2} - 4 - 5 x = 20$

Step 6

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

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Step 6.1

Add $4$ to both sides of the equation.

$x^{2} - 5 x = 20 + 4$

Step 6.2

Add $20$ and $4$ .

$x^{2} - 5 x = 24$

Step 7

Subtract $24$ from both sides of the equation.

$x^{2} - 5 x - 24 = 0$

Step 8

Factor $x^{2} - 5 x - 24$ using the AC method.

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Step 8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 8.2

Write the factored form using these integers.

$(x - 8) (x + 3) = 0$

Step 9

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 8 = 0$

$x + 3 = 0$

Step 10

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 10.2

Add $8$ to both sides of the equation.

$x = 8$

Step 11

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 11.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 12

The final solution is all the values that make $(x - 8) (x + 3) = 0$ true.

$x = 8, - 3$