Precalculus Examples

$e^{1 - 4 x} = 2$

Step 1

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

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

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{1 - 4 x})$ by moving $1 - 4 x$ outside the logarithm.

$(1 - 4 x) \ln (e) = \ln (2)$

Step 2.2

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

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

Step 2.3

Multiply $1 - 4 x$ by $1$ .

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

Step 3

Subtract $1$ from both sides of the equation.

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

Step 4

Divide each term in $- 4 x = \ln (2) - 1$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 x = \ln (2) - 1$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{\ln (2)}{- 4} + \frac{- 1}{- 4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{\ln (2)}{- 4} + \frac{- 1}{- 4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{- 4} + \frac{- 1}{- 4}$

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 in front of the fraction.

$x = - \frac{\ln (2)}{4} + \frac{- 1}{- 4}$

Step 4.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{\ln (2)}{4} + \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (2)}{4} + \frac{1}{4}$

Decimal Form:

$x = 0.07671320 \dots$