Precalculus Examples

$e^{1 - x} = 4$

Step 1

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

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

Step 2

Expand the left side.

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Step 2.1

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

$(1 - x) \ln (e) = \ln (4)$

Step 2.2

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

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

Step 2.3

Multiply $1 - x$ by $1$ .

$1 - x = \ln (4)$

Step 3

Subtract $1$ from both sides of the equation.

$- x = \ln (4) - 1$

Step 4

Divide each term in $- x = \ln (4) - 1$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = \ln (4) - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (4)}{- 1} + \frac{- 1}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (4)}{- 1} + \frac{- 1}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{- 1} + \frac{- 1}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\ln (4)}{- 1}$ .

$x = - 1 \cdot \ln (4) + \frac{- 1}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \ln (4)$ as $- \ln (4)$ .

$x = - \ln (4) + \frac{- 1}{- 1}$

Step 4.3.1.3

Divide $- 1$ by $- 1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.38629436 \dots$