Precalculus Examples

$e^{- 2 x} = \frac{1}{3}$

Step 1

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

$\ln (e^{- 2 x}) = \ln (\frac{1}{3})$

Step 2

Expand the left side.

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Step 2.1

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

$- 2 x \ln (e) = \ln (\frac{1}{3})$

Step 2.2

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

$- 2 x \cdot 1 = \ln (\frac{1}{3})$

Step 2.3

Multiply $- 2$ by $1$ .

$- 2 x = \ln (\frac{1}{3})$

Step 3

Divide each term in $- 2 x = \ln (\frac{1}{3})$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 x = \ln (\frac{1}{3})$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{\ln (\frac{1}{3})}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{\ln (\frac{1}{3})}{- 2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{3})}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{3})}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{1}{3})}{2}$

Decimal Form:

$x = 0.54930614 \dots$