Algebra Examples

$\ln (x - 5) + \ln (x + 2) = \ln (x - 14)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((x - 5) (x + 2)) = \ln (x - 14)$

Step 1.2

Expand $(x - 5) (x + 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (x (x + 2) - 5 (x + 2)) = \ln (x - 14)$

Step 1.2.2

Apply the distributive property.

$\ln (x \cdot x + x \cdot 2 - 5 (x + 2)) = \ln (x - 14)$

Step 1.2.3

Apply the distributive property.

$\ln (x \cdot x + x \cdot 2 - 5 x - 5 \cdot 2) = \ln (x - 14)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot 2 - 5 x - 5 \cdot 2) = \ln (x - 14)$

Step 1.3.1.2

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

$\ln (x^{2} + 2 \cdot x - 5 x - 5 \cdot 2) = \ln (x - 14)$

Step 1.3.1.3

Multiply $- 5$ by $2$ .

$\ln (x^{2} + 2 x - 5 x - 10) = \ln (x - 14)$

Step 1.3.2

Subtract $5 x$ from $2 x$ .

$\ln (x^{2} - 3 x - 10) = \ln (x - 14)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} - 3 x - 10 = x - 14$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$x^{2} - 3 x - 10 - x = - 14$

Step 3.1.2

Subtract $x$ from $- 3 x$ .

$x^{2} - 4 x - 10 = - 14$

Step 3.2

Add $14$ to both sides of the equation.

$x^{2} - 4 x - 10 + 14 = 0$

Step 3.3

Add $- 10$ and $14$ .

$x^{2} - 4 x + 4 = 0$

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

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

$x^{2} - 4 x + 2^{2} = 0$

Step 3.4.2

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

$4 x = 2 \cdot x \cdot 2$

Step 3.4.3

Rewrite the polynomial.

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

Step 3.4.4

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

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

Step 3.5

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.6

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Exclude the solutions that do not make $\ln (x - 5) + \ln (x + 2) = \ln (x - 14)$ true.

$No solution$