Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Cross multiply to remove the fraction.

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

Step 4

Simplify $e (x - 1)$ .

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Step 4.1

Simplify by multiplying through.

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Step 4.1.1

Apply the distributive property.

$x + 2 = e x + e \cdot - 1$

Step 4.1.2

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

$x + 2 = e x - 1 \cdot e$

Step 4.2

Rewrite $- 1 e$ as $- e$ .

$x + 2 = e x - e$

Step 5

Subtract $e x$ from both sides of the equation.

$x + 2 - e x = - e$

Step 6

Subtract $2$ from both sides of the equation.

$x - e x = - e - 2$

Step 7

Factor $x$ out of $x - e x$ .

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Step 7.1

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

$x \cdot 1 - e x = - e - 2$

Step 7.2

Factor $x$ out of $- e x$ .

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

Step 7.3

Factor $x$ out of $x \cdot 1 + x (- e)$ .

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

Step 8

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

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Step 8.1

Divide each term in $x (1 - e) = - e - 2$ by $1 - e$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $1 - e$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

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

Step 8.3.2

Factor $- 1$ out of $- e$ .

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

Step 8.3.3

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 8.3.4

Factor $- 1$ out of $- (e) - 1 (2)$ .

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

Step 8.3.5

Simplify the expression.

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Step 8.3.5.1

Rewrite $- (e + 2)$ as $- 1 (e + 2)$ .

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

Step 8.3.5.2

Move the negative in front of the fraction.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.74593012 \dots$