Precalculus Examples

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

Step 1

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

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

Step 2

Expand the left side.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $x - 1$ by $1$ .

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

Step 3

Expand the right side.

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Step 3.1

Expand $\ln ({(\frac{1}{e^{2}})}^{x + 5})$ by moving $x + 5$ outside the logarithm.

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

Step 3.2

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

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

Step 3.3

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

$x - 1 = (x + 5) (\ln (1) - (2 \ln (e)))$

Step 3.4

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

$x - 1 = (x + 5) (0 - (2 \ln (e)))$

Step 3.5

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

$x - 1 = (x + 5) (0 - (2 \cdot 1))$

Step 3.6

Multiply $2$ by $1$ .

$x - 1 = (x + 5) (0 - 1 \cdot 2)$

Step 3.7

Multiply $- 1$ by $2$ .

$x - 1 = (x + 5) (0 - 2)$

Step 3.8

Subtract $2$ from $0$ .

$x - 1 = (x + 5) \cdot - 2$

Step 4

Simplify the right side.

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Step 4.1

Simplify $(x + 5) \cdot - 2$ .

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Step 4.1.1

Apply the distributive property.

$x - 1 = x \cdot - 2 + 5 \cdot - 2$

Step 4.1.2

Simplify the expression.

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Step 4.1.2.1

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

$x - 1 = - 2 \cdot x + 5 \cdot - 2$

Step 4.1.2.2

Multiply $5$ by $- 2$ .

$x - 1 = - 2 x - 10$

Step 5

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

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Step 5.1

Add $2 x$ to both sides of the equation.

$x - 1 + 2 x = - 10$

Step 5.2

Add $x$ and $2 x$ .

$3 x - 1 = - 10$

Step 6

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

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Step 6.1

Add $1$ to both sides of the equation.

$3 x = - 10 + 1$

Step 6.2

Add $- 10$ and $1$ .

$3 x = - 9$

Step 7

Divide each term in $3 x = - 9$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 x = - 9$ by $3$ .

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 9$ by $3$ .

$x = - 3$