Precalculus Examples

$3 + 2 e^{- x} = 6$

Step 1

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

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Step 1.1

Subtract $3$ from both sides of the equation.

$2 e^{- x} = 6 - 3$

Step 1.2

Subtract $3$ from $6$ .

$2 e^{- x} = 3$

Step 2

Divide each term in $2 e^{- x} = 3$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 e^{- x} = 3$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $e^{- x}$ by $1$ .

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

Step 3

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

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

Step 4

Expand the left side.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $- 1$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative one from the denominator of $\frac{\ln (\frac{3}{2})}{- 1}$ .

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

Step 5.3.2

Rewrite $- 1 \cdot \ln (\frac{3}{2})$ as $- \ln (\frac{3}{2})$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.40546510 \dots$