Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Cross multiply to remove the fraction.

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

Step 4

Multiply $e^{2}$ by $x$ .

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

Step 5

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

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

Step 6

Subtract $2$ from both sides of the equation.

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

Step 7

Factor the left side of the equation.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

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

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

Step 7.2

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

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

Step 7.3

Factor.

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Step 7.3.1

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 $b = e$ .

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

Step 7.3.2

Remove unnecessary parentheses.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Simplify the denominator.

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Step 8.2.1.1

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

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

Step 8.2.1.2

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 $b = e$ .

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

Step 8.2.2

Reduce the expression by cancelling the common factors.

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Step 8.2.2.1

Cancel the common factor of $1 + e$ .

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Rewrite the expression.

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

Step 8.2.2.2

Cancel the common factor of $1 - e$ .

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Step 8.2.2.2.1

Cancel the common factor.

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

Step 8.2.2.2.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify the denominator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

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 $b = e$ .

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

Step 8.3.2

Move the negative in front of the fraction.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.31303528 \dots$