Precalculus Examples

$e^{x} - e^{- x} - 1 = 0$

Step 1

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

$e^{x} - {(e^{x})}^{- 1} - 1 = 0$

Step 2

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

$u - u^{- 1} - 1 = 0$

Step 3

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

$u - \frac{1}{u} - 1 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$u$

Step 4.2

Multiply each term in $u - \frac{1}{u} - 1 = 0$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u - \frac{1}{u} - 1 = 0$ by $u$ .

$u \cdot u - \frac{1}{u} u - u = 0 u$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} - \frac{1}{u} u - u = 0 u$

Step 4.2.2.1.2

Cancel the common factor of $u$ .

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Step 4.2.2.1.2.1

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

$u^{2} + \frac{- 1}{u} u - u = 0 u$

Step 4.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 1}{u} u - u = 0 u$

Step 4.2.2.1.2.3

Rewrite the expression.

$u^{2} - 1 - u = 0 u$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $0$ by $u$ .

$u^{2} - 1 - u = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 4.3.3

Simplify.

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Step 4.3.3.1

Simplify the numerator.

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Step 4.3.3.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 4.3.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.3.3.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 1}}{2 \cdot 1}$

Step 4.3.3.1.2.2

Multiply $- 4$ by $- 1$ .

$u = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 4.3.3.1.3

Add $1$ and $4$ .

$u = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 4.3.3.2

Multiply $2$ by $1$ .

$u = \frac{1 \pm \sqrt{5}}{2}$

Step 4.3.4

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 5

Substitute $\frac{1 + \sqrt{5}}{2}$ for $u$ in $u = e^{x}$ .

$\frac{1 + \sqrt{5}}{2} = e^{x}$

Step 6

Solve $\frac{1 + \sqrt{5}}{2} = e^{x}$ .

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Step 6.1

Rewrite the equation as $e^{x} = \frac{1 + \sqrt{5}}{2}$ .

$e^{x} = \frac{1 + \sqrt{5}}{2}$

Step 6.2

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

$\ln (e^{x}) = \ln (\frac{1 + \sqrt{5}}{2})$

Step 6.3

Expand the left side.

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Step 6.3.1

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

$x \ln (e) = \ln (\frac{1 + \sqrt{5}}{2})$

Step 6.3.2

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

$x \cdot 1 = \ln (\frac{1 + \sqrt{5}}{2})$

Step 6.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{1 + \sqrt{5}}{2})$

Step 7

Substitute $\frac{1 - \sqrt{5}}{2}$ for $u$ in $u = e^{x}$ .

$\frac{1 - \sqrt{5}}{2} = e^{x}$

Step 8

Solve $\frac{1 - \sqrt{5}}{2} = e^{x}$ .

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Step 8.1

Rewrite the equation as $e^{x} = \frac{1 - \sqrt{5}}{2}$ .

$e^{x} = \frac{1 - \sqrt{5}}{2}$

Step 8.2

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

$\ln (e^{x}) = \ln (\frac{1 - \sqrt{5}}{2})$

Step 8.3

The equation cannot be solved because $\ln (\frac{1 - \sqrt{5}}{2})$ is undefined.

Undefined

Step 8.4

There is no solution for $e^{x} = \frac{1 - \sqrt{5}}{2}$

No solution

Step 9

List the solutions that makes the equation true.

$x = \ln (\frac{1 + \sqrt{5}}{2})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{1 + \sqrt{5}}{2})$

Decimal Form:

$x = 0.48121182 \dots$