Finite Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Cross multiply to remove the fraction.

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

Step 5

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

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine $\frac{1}{e}$ and $x$ .

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

Step 5.4

Combine $\frac{1}{e}$ and $- 1$ .

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

Step 5.5

Move the negative in front of the fraction.

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

Step 6

Subtract $\frac{x}{e}$ from both sides of the equation.

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

Step 7

Subtract $1$ from both sides of the equation.

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

Step 8

Factor $x$ out of $x - \frac{x}{e}$ .

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Step 8.1

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

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

Step 8.2

Factor $x$ out of $- \frac{x}{e}$ .

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

Step 8.3

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

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

Step 9

Divide each term in $x (1 - \frac{1}{e}) = - \frac{1}{e} - 1$ by $1 - \frac{1}{e}$ and simplify.

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Step 9.1

Divide each term in $x (1 - \frac{1}{e}) = - \frac{1}{e} - 1$ by $1 - \frac{1}{e}$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $1 - \frac{1}{e}$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Combine the numerators over the common denominator.

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

Step 9.3.2

Multiply the numerator and denominator of the fraction by $e$ .

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Step 9.3.2.1

Multiply $\frac{- \frac{1}{e} - 1}{1 - \frac{1}{e}}$ by $\frac{e}{e}$ .

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

Step 9.3.2.2

Combine.

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

Step 9.3.3

Apply the distributive property.

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

Step 9.3.4

Simplify by cancelling.

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Step 9.3.4.1

Cancel the common factor of $e$ .

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Step 9.3.4.1.1

Move the leading negative in $- \frac{1}{e}$ into the numerator.

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

Step 9.3.4.1.2

Cancel the common factor.

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

Step 9.3.4.1.3

Rewrite the expression.

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

Step 9.3.4.2

Cancel the common factor of $e$ .

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Step 9.3.4.2.1

Move the leading negative in $- \frac{1}{e}$ into the numerator.

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

Step 9.3.4.2.2

Cancel the common factor.

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

Step 9.3.4.2.3

Rewrite the expression.

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

Step 9.3.5

Simplify the numerator.

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Step 9.3.5.1

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

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

Step 9.3.5.2

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

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

Step 9.3.6

Simplify with factoring out.

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Step 9.3.6.1

Multiply $e$ by $1$ .

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

Step 9.3.6.2

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

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

Step 9.3.6.3

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

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

Step 9.3.6.4

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

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

Step 9.3.6.5

Move the negative in front of the fraction.

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

Step 10

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

$No solution$