Precalculus Examples

$\ln (x - 2) - \ln (x + 2) = \ln (x - 1) - \ln (2 x + 1)$

Step 1

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

$\ln (\frac{x - 2}{x + 2}) = \ln (x - 1) - \ln (2 x + 1)$

Step 2

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

$\ln (\frac{x - 2}{x + 2}) = \ln (\frac{x - 1}{2 x + 1})$

Step 3

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

$\frac{x - 2}{x + 2} = \frac{x - 1}{2 x + 1}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Simplify $(x - 2) (2 x + 1)$ .

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Step 4.2.1.1

Rewrite.

$0 + 0 + (x - 2) (2 x + 1) = (x + 2) (x - 1)$

Step 4.2.1.2

Simplify by adding zeros.

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

Step 4.2.1.3

Expand $(x - 2) (2 x + 1)$ using the FOIL Method.

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Step 4.2.1.3.1

Apply the distributive property.

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

Step 4.2.1.3.2

Apply the distributive property.

$x (2 x) + x \cdot 1 - 2 (2 x + 1) = (x + 2) (x - 1)$

Step 4.2.1.3.3

Apply the distributive property.

$x (2 x) + x \cdot 1 - 2 (2 x) - 2 \cdot 1 = (x + 2) (x - 1)$

Step 4.2.1.4

Simplify and combine like terms.

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Step 4.2.1.4.1

Simplify each term.

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Step 4.2.1.4.1.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot 1 - 2 (2 x) - 2 \cdot 1 = (x + 2) (x - 1)$

Step 4.2.1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.1.4.1.2.1

Move $x$ .

$2 (x \cdot x) + x \cdot 1 - 2 (2 x) - 2 \cdot 1 = (x + 2) (x - 1)$

Step 4.2.1.4.1.2.2

Multiply $x$ by $x$ .

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

Step 4.2.1.4.1.3

Multiply $x$ by $1$ .

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

Step 4.2.1.4.1.4

Multiply $2$ by $- 2$ .

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

Step 4.2.1.4.1.5

Multiply $- 2$ by $1$ .

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

Step 4.2.1.4.2

Subtract $4 x$ from $x$ .

$2 x^{2} - 3 x - 2 = (x + 2) (x - 1)$

Step 4.2.2

Simplify $(x + 2) (x - 1)$ .

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Step 4.2.2.1

Expand $(x + 2) (x - 1)$ using the FOIL Method.

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Step 4.2.2.1.1

Apply the distributive property.

$2 x^{2} - 3 x - 2 = x (x - 1) + 2 (x - 1)$

Step 4.2.2.1.2

Apply the distributive property.

$2 x^{2} - 3 x - 2 = x \cdot x + x \cdot - 1 + 2 (x - 1)$

Step 4.2.2.1.3

Apply the distributive property.

$2 x^{2} - 3 x - 2 = x \cdot x + x \cdot - 1 + 2 x + 2 \cdot - 1$

Step 4.2.2.2

Simplify and combine like terms.

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Step 4.2.2.2.1

Simplify each term.

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Step 4.2.2.2.1.1

Multiply $x$ by $x$ .

$2 x^{2} - 3 x - 2 = x^{2} + x \cdot - 1 + 2 x + 2 \cdot - 1$

Step 4.2.2.2.1.2

Move $- 1$ to the left of $x$ .

$2 x^{2} - 3 x - 2 = x^{2} - 1 \cdot x + 2 x + 2 \cdot - 1$

Step 4.2.2.2.1.3

Rewrite $- 1 x$ as $- x$ .

$2 x^{2} - 3 x - 2 = x^{2} - x + 2 x + 2 \cdot - 1$

Step 4.2.2.2.1.4

Multiply $2$ by $- 1$ .

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

Step 4.2.2.2.2

Add $- x$ and $2 x$ .

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

Step 4.2.3

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

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Step 4.2.3.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 4.2.3.2

Subtract $x$ from both sides of the equation.

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

Step 4.2.3.3

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 4.2.3.4

Subtract $x$ from $- 3 x$ .

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

Step 4.2.4

Add $2$ to both sides of the equation.

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

Step 4.2.5

Combine the opposite terms in $x^{2} - 4 x - 2 + 2$ .

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Step 4.2.5.1

Add $- 2$ and $2$ .

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

Step 4.2.5.2

Add $x^{2} - 4 x$ and $0$ .

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

Step 4.2.6

Factor $x$ out of $x^{2} - 4 x$ .

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Step 4.2.6.1

Factor $x$ out of $x^{2}$ .

$x \cdot x - 4 x = 0$

Step 4.2.6.2

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Step 4.2.6.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

Step 4.2.7

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

$x = 0$

$x - 4 = 0$

Step 4.2.8

Set $x$ equal to $0$ .

$x = 0$

Step 4.2.9

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

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Step 4.2.9.1

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

$x - 4 = 0$

Step 4.2.9.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.2.10

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 5

Exclude the solutions that do not make $\ln (x - 2) - \ln (x + 2) = \ln (x - 1) - \ln (2 x + 1)$ true.

$x = 4$