Calculus Examples

$3 = 2 + 5 e^{- 4 x}$

Step 1

Rewrite the equation as $2 + 5 e^{- 4 x} = 3$ .

$2 + 5 e^{- 4 x} = 3$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the equation.

$5 e^{- 4 x} = 3 - 2$

Step 2.2

Subtract $2$ from $3$ .

$5 e^{- 4 x} = 1$

Step 3

Divide each term in $5 e^{- 4 x} = 1$ by $5$ and simplify.

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Step 3.1

Divide each term in $5 e^{- 4 x} = 1$ by $5$ .

$\frac{5 e^{- 4 x}}{5} = \frac{1}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 e^{- 4 x}}{5} = \frac{1}{5}$

Step 3.2.1.2

Divide $e^{- 4 x}$ by $1$ .

$e^{- 4 x} = \frac{1}{5}$

Step 4

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

$\ln (e^{- 4 x}) = \ln (\frac{1}{5})$

Step 5

Expand the left side.

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Step 5.1

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

$- 4 x \ln (e) = \ln (\frac{1}{5})$

Step 5.2

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

$- 4 x \cdot 1 = \ln (\frac{1}{5})$

Step 5.3

Multiply $- 4$ by $1$ .

$- 4 x = \ln (\frac{1}{5})$

Step 6

Divide each term in $- 4 x = \ln (\frac{1}{5})$ by $- 4$ and simplify.

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Step 6.1

Divide each term in $- 4 x = \ln (\frac{1}{5})$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{1}{5})}{- 4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{1}{5})}{- 4}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{5})}{- 4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{5})}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{1}{5})}{4}$

Decimal Form:

$x = 0.40235947 \dots$