Algebra Examples

$\log_{2} (x^{2} - 7) - \log_{2} (x + 5) = 0$

Step 1

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

$\log_{2} (\frac{x^{2} - 7}{x + 5}) = 0$

Step 2

Rewrite $\log_{2} (\frac{x^{2} - 7}{x + 5}) = 0$ 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 $b^{y} = x$ .

$2^{0} = \frac{x^{2} - 7}{x + 5}$

Step 3

Cross multiply to remove the fraction.

$x^{2} - 7 = 2^{0} (x + 5)$

Step 4

Simplify $2^{0} (x + 5)$ .

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Step 4.1

Anything raised to $0$ is $1$ .

$x^{2} - 7 = 1 (x + 5)$

Step 4.2

Multiply $x + 5$ by $1$ .

$x^{2} - 7 = x + 5$

Step 5

Subtract $x$ from both sides of the equation.

$x^{2} - 7 - x = 5$

Step 6

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

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Step 6.1

Add $7$ to both sides of the equation.

$x^{2} - x = 5 + 7$

Step 6.2

Add $5$ and $7$ .

$x^{2} - x = 12$

Step 7

Subtract $12$ from both sides of the equation.

$x^{2} - x - 12 = 0$

Step 8

Factor $x^{2} - x - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 8.2

Write the factored form using these integers.

$(x - 4) (x + 3) = 0$

Step 9

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

$x - 4 = 0$

$x + 3 = 0$

Step 10

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 10.2

Add $4$ to both sides of the equation.

$x = 4$

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 $(x - 4) (x + 3) = 0$ true.

$x = 4, - 3$