Algebra Examples

$\ln (x - 3) + \ln (x + 2) = \ln (x - 7)$

Step 1

Subtract $\ln (x - 7)$ from both sides of the equation.

$\ln (x - 3) + \ln (x + 2) - \ln (x - 7) = 0$

Step 2

Simplify $\ln (x - 3) + \ln (x + 2) - \ln (x - 7)$ .

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Step 2.1

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

$\ln ((x - 3) (x + 2)) - \ln (x - 7) = 0$

Step 2.2

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

$\ln (\frac{(x - 3) (x + 2)}{x - 7}) = 0$

Step 3

Rewrite $\ln (\frac{(x - 3) (x + 2)}{x - 7}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{(x - 3) (x + 2)}{x - 7}$

Step 4

Cross multiply to remove the fraction.

$(x - 3) (x + 2) = e^{0} (x - 7)$

Step 5

Simplify $e^{0} (x - 7)$ .

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Step 5.1

Anything raised to $0$ is $1$ .

$(x - 3) (x + 2) = 1 (x - 7)$

Step 5.2

Multiply $x - 7$ by $1$ .

$(x - 3) (x + 2) = x - 7$

Step 6

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

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Step 6.1

Subtract $x$ from both sides of the equation.

$(x - 3) (x + 2) - x = - 7$

Step 6.2

Simplify each term.

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Step 6.2.1

Expand $(x - 3) (x + 2)$ using the FOIL Method.

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Step 6.2.1.1

Apply the distributive property.

$x (x + 2) - 3 (x + 2) - x = - 7$

Step 6.2.1.2

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 (x + 2) - x = - 7$

Step 6.2.1.3

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2 - x = - 7$

Step 6.2.2

Simplify and combine like terms.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 - 3 x - 3 \cdot 2 - x = - 7$

Step 6.2.2.1.2

Move $2$ to the left of $x$ .

$x^{2} + 2 \cdot x - 3 x - 3 \cdot 2 - x = - 7$

Step 6.2.2.1.3

Multiply $- 3$ by $2$ .

$x^{2} + 2 x - 3 x - 6 - x = - 7$

Step 6.2.2.2

Subtract $3 x$ from $2 x$ .

$x^{2} - x - 6 - x = - 7$

Step 6.3

Subtract $x$ from $- x$ .

$x^{2} - 2 x - 6 = - 7$

Step 7

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

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Step 7.1

Add $6$ to both sides of the equation.

$x^{2} - 2 x = - 7 + 6$

Step 7.2

Add $- 7$ and $6$ .

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

Step 8

Add $1$ to both sides of the equation.

$x^{2} - 2 x + 1 = 0$

Step 9

Factor using the perfect square rule.

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Step 9.1

Rewrite $1$ as $1^{2}$ .

$x^{2} - 2 x + 1^{2} = 0$

Step 9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 9.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 1 + 1^{2} = 0$

Step 9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

${(x - 1)}^{2} = 0$

Step 10

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 11

Add $1$ to both sides of the equation.

$x = 1$