Precalculus Examples

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

Step 1

Move all the terms containing a logarithm to the left side of the equation.

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

Step 2

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

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

Step 3

Rewrite $\ln (\frac{x + 1}{x - 1}) = 2$ 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 $b^{y} = x$ .

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

Step 4

Cross multiply to remove the fraction.

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

Step 5

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

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Step 5.1

Simplify by multiplying through.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Move $- 1$ to the left of $e^{2}$ .

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

Step 5.2

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

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

Step 6

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

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

Step 7

Subtract $1$ from both sides of the equation.

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

Step 8

Factor the left side of the equation.

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Step 8.1

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

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Step 8.1.1

Raise $x$ to the power of $1$ .

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

Step 8.1.2

Factor $x$ out of $x^{1}$ .

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

Step 8.1.3

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

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

Step 8.1.4

Factor $x$ out of $x \cdot 1 + x (- e^{2})$ .

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

Step 8.2

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

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

Step 8.3

Factor.

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Step 8.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

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

Step 8.3.2

Remove unnecessary parentheses.

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

Step 9

Divide each term in $x (1 + e) (1 - e) = - e^{2} - 1$ by $1 - e^{2}$ and simplify.

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Step 9.1

Divide each term in $x (1 + e) (1 - e) = - e^{2} - 1$ by $1 - e^{2}$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Simplify the denominator.

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Step 9.2.1.1

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

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

Step 9.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

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

Step 9.2.2

Reduce the expression by cancelling the common factors.

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Step 9.2.2.1

Cancel the common factor of $1 + e$ .

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Step 9.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.1.2

Rewrite the expression.

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

Step 9.2.2.2

Cancel the common factor of $1 - e$ .

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Step 9.2.2.2.1

Cancel the common factor.

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

Step 9.2.2.2.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Combine the numerators over the common denominator.

$x = \frac{- e^{2} - 1}{1 - e^{2}}$

Step 9.3.2

Simplify the denominator.

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Step 9.3.2.1

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

$x = \frac{- e^{2} - 1}{1^{2} - e^{2}}$

Step 9.3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

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

Step 9.3.3

Simplify with factoring out.

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Step 9.3.3.1

Factor $- 1$ out of $- e^{2}$ .

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

Step 9.3.3.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 9.3.3.3

Factor $- 1$ out of $- (e^{2}) - 1 (1)$ .

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

Step 9.3.3.4

Simplify the expression.

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Step 9.3.3.4.1

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

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

Step 9.3.3.4.2

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.31303528 \dots$