Precalculus Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\ln (x - 1), 1$

Step 1.2

The LCM of one and any expression is the expression.

$\ln (x - 1)$

Step 2

Multiply each term in $\frac{\ln (x + 1)}{\ln (x - 1)} = 2$ by $\ln (x - 1)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{\ln (x + 1)}{\ln (x - 1)} = 2$ by $\ln (x - 1)$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\ln (x - 1)$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $2 \ln (x - 1)$ by moving $2$ inside the logarithm.

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

Step 3

Solve the equation.

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Step 3.1

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

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Simplify ${(x - 1)}^{2}$ .

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Step 3.2.1.1

Rewrite.

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

Step 3.2.1.2

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

Apply the distributive property.

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

Step 3.2.1.3.2

Apply the distributive property.

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

Step 3.2.1.3.3

Apply the distributive property.

$x + 1 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 3.2.1.4

Simplify and combine like terms.

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Step 3.2.1.4.1

Simplify each term.

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Step 3.2.1.4.1.1

Multiply $x$ by $x$ .

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

Step 3.2.1.4.1.2

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

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

Step 3.2.1.4.1.3

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

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

Step 3.2.1.4.1.4

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

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

Step 3.2.1.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.4.2

Subtract $x$ from $- x$ .

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

Step 3.2.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3.2.3

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

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Step 3.2.3.1

Subtract $x$ from both sides of the equation.

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

Step 3.2.3.2

Subtract $x$ from $- 2 x$ .

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

Step 3.2.4

Subtract $1$ from both sides of the equation.

$x^{2} - 3 x + 1 - 1 = 0$

Step 3.2.5

Combine the opposite terms in $x^{2} - 3 x + 1 - 1$ .

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Step 3.2.5.1

Subtract $1$ from $1$ .

$x^{2} - 3 x + 0 = 0$

Step 3.2.5.2

Add $x^{2} - 3 x$ and $0$ .

$x^{2} - 3 x = 0$

Step 3.2.6

Factor $x$ out of $x^{2} - 3 x$ .

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Step 3.2.6.1

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

$x \cdot x - 3 x = 0$

Step 3.2.6.2

Factor $x$ out of $- 3 x$ .

$x \cdot x + x \cdot - 3 = 0$

Step 3.2.6.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$x (x - 3) = 0$

Step 3.2.7

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

$x = 0$

$x - 3 = 0$

Step 3.2.8

Set $x$ equal to $0$ .

$x = 0$

Step 3.2.9

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

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Step 3.2.9.1

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

$x - 3 = 0$

Step 3.2.9.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.2.10

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

$x = 0, 3$

Step 4

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

$x = 3$