Precalculus Examples

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

Step 1

Rewrite the equation as ${(\frac{1}{e^{2}})}^{x + 1} = {(\frac{1}{e})}^{- 1}$ .

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

Step 2

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

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

Step 3

Expand the left side.

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Step 3.1

Expand $\ln ({(\frac{1}{e^{2}})}^{x + 1})$ by moving $x + 1$ outside the logarithm.

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

Step 3.2

Rewrite $\ln (\frac{1}{e^{2}})$ as $\ln (1) - \ln (e^{2})$ .

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

Step 3.3

Expand $\ln (e^{2})$ by moving $2$ outside the logarithm.

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

Step 3.4

The natural logarithm of $1$ is $0$ .

$(x + 1) (0 - (2 \ln (e))) = \ln ({(\frac{1}{e})}^{- 1})$

Step 3.5

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

$(x + 1) (0 - (2 \cdot 1)) = \ln ({(\frac{1}{e})}^{- 1})$

Step 3.6

Multiply $2$ by $1$ .

$(x + 1) (0 - 1 \cdot 2) = \ln ({(\frac{1}{e})}^{- 1})$

Step 3.7

Multiply $- 1$ by $2$ .

$(x + 1) (0 - 2) = \ln ({(\frac{1}{e})}^{- 1})$

Step 3.8

Subtract $2$ from $0$ .

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

Step 4

Expand the right side.

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Step 4.1

Expand $\ln ({(\frac{1}{e})}^{- 1})$ by moving $- 1$ outside the logarithm.

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

Step 4.2

Rewrite $\frac{1}{e}$ as $1 e^{- 1}$ .

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

Step 4.3

Multiply $e^{- 1}$ by $1$ .

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

Step 4.4

Expand $\ln (e^{- 1})$ by moving $- 1$ outside the logarithm.

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

Step 4.5

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

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

Step 4.6

Multiply $- 1$ by $1$ .

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

Step 4.7

Multiply $- 1$ by $- 1$ .

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

Step 5

Simplify the left side.

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Step 5.1

Simplify $(x + 1) \cdot - 2$ .

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Simplify the expression.

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Step 5.1.2.1

Move $- 2$ to the left of $x$ .

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

Step 5.1.2.2

Multiply $- 2$ by $1$ .

$- 2 x - 2 = 1$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Add $2$ to both sides of the equation.

$- 2 x = 1 + 2$

Step 6.2

Add $1$ and $2$ .

$- 2 x = 3$

Step 7

Divide each term in $- 2 x = 3$ by $- 2$ and simplify.

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Step 7.1

Divide each term in $- 2 x = 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{- 2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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