Precalculus Examples

$\frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} = 3$

Step 1

Multiply both sides by $e^{x} - e^{- x}$ .

$\frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} (e^{x} - e^{- x}) = 3 (e^{x} - e^{- x})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $e^{x} - e^{- x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} (e^{x} - e^{- x}) = 3 (e^{x} - e^{- x})$

Step 2.1.1.2

Rewrite the expression.

$e^{x} + e^{- x} = 3 (e^{x} - e^{- x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $3 (e^{x} - e^{- x})$ .

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Step 2.2.1.1

Apply the distributive property.

$e^{x} + e^{- x} = 3 e^{x} + 3 (- e^{- x})$

Step 2.2.1.2

Multiply $- 1$ by $3$ .

$e^{x} + e^{- x} = 3 e^{x} - 3 e^{- x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite $e^{- x}$ as exponentiation.

$e^{x} + {(e^{x})}^{- 1} = 3 e^{x} - 3 e^{- x}$

Step 3.2

Substitute $u$ for $e^{x}$ .

$u + u^{- 1} = 3 e^{x} - 3 e^{- x}$

Step 3.3

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

$u + \frac{1}{u} = 3 e^{x} - 3 e^{- x}$

Step 3.4

Rewrite $e^{- x}$ as exponentiation.

$u + \frac{1}{u} = 3 e^{x} - 3 {(e^{x})}^{- 1} = 3 e^{x} - 3 e^{- x}$

Step 3.5

Substitute $u$ for $e^{x}$ .

$u + \frac{1}{u} = 3 u - 3 u^{- 1} = 3 e^{x} - 3 e^{- x}$

Step 3.6

Simplify each term.

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Step 3.6.1

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

$u + \frac{1}{u} = 3 u - 3 \frac{1}{u} = 3 e^{x} - 3 e^{- x}$

Step 3.6.2

Combine $- 3$ and $\frac{1}{u}$ .

$u + \frac{1}{u} = 3 u + \frac{- 3}{u} = 3 e^{x} - 3 e^{- x}$

Step 3.6.3

Move the negative in front of the fraction.

$u + \frac{1}{u} = 3 u - \frac{3}{u} = 3 e^{x} - 3 e^{- x}$

Step 3.7

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

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Step 3.7.1

Subtract $3 e^{x}$ from both sides of the equation.

$e^{x} + e^{- x} - 3 e^{x} = - 3 e^{- x}$

Step 3.7.2

Add $3 e^{- x}$ to both sides of the equation.

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

Step 3.7.3

Subtract $3 e^{x}$ from $e^{x}$ .

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

Step 3.7.4

Add $e^{- x}$ and $3 e^{- x}$ .

$4 e^{- x} - 2 e^{x} = 0$

Step 3.8

Move $- 2 e^{x}$ to the right side of the equation by adding it to both sides.

$4 e^{- x} = 2 e^{x}$

Step 3.9

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4 e^{- x}) = \ln (2 e^{x})$

Step 3.10

Expand the left side.

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Step 3.10.1

Rewrite $\ln (4 e^{- x})$ as $\ln (4) + \ln (e^{- x})$ .

$\ln (4) + \ln (e^{- x}) = \ln (2 e^{x})$

Step 3.10.2

Expand $\ln (e^{- x})$ by moving $- x$ outside the logarithm.

$\ln (4) - x \ln (e) = \ln (2 e^{x})$

Step 3.10.3

The natural logarithm of $e$ is $1$ .

$\ln (4) - x \cdot 1 = \ln (2 e^{x})$

Step 3.10.4

Multiply $- 1$ by $1$ .

$\ln (4) - x = \ln (2 e^{x})$

Step 3.11

Expand the right side.

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Step 3.11.1

Rewrite $\ln (2 e^{x})$ as $\ln (2) + \ln (e^{x})$ .

$\ln (4) - x = \ln (2) + \ln (e^{x})$

Step 3.11.2

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$\ln (4) - x = \ln (2) + x \ln (e)$

Step 3.11.3

The natural logarithm of $e$ is $1$ .

$\ln (4) - x = \ln (2) + x \cdot 1$

Step 3.11.4

Multiply $x$ by $1$ .

$\ln (4) - x = \ln (2) + x$

Step 3.12

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

$\ln (4) - \ln (2) = x + x$

Step 3.13

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

$\ln (\frac{4}{2}) = x + x$

Step 3.14

Divide $4$ by $2$ .

$\ln (2) = x + x$

Step 3.15

Add $x$ and $x$ .

$\ln (2) = 2 x$

Step 3.16

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

$2 x = \ln (2)$

Step 3.17

Divide each term in $2 x = \ln (2)$ by $2$ and simplify.

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Step 3.17.1

Divide each term in $2 x = \ln (2)$ by $2$ .

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

Step 3.17.2

Simplify the left side.

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Step 3.17.2.1

Cancel the common factor of $2$ .

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Step 3.17.2.1.1

Cancel the common factor.

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

Step 3.17.2.1.2

Divide $x$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.34657359 \dots$