Calculus Examples

$\frac{90}{1 + e^{- x}} = 4$

Step 1

Multiply both sides by $1 + e^{- x}$ .

$\frac{90}{1 + e^{- x}} (1 + e^{- x}) = 4 (1 + e^{- x})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $1 + e^{- x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{90}{1 + e^{- x}} (1 + e^{- x}) = 4 (1 + e^{- x})$

Step 2.1.1.2

Rewrite the expression.

$90 = 4 (1 + e^{- x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $4 (1 + e^{- x})$ .

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Step 2.2.1.1

Apply the distributive property.

$90 = 4 \cdot 1 + 4 e^{- x}$

Step 2.2.1.2

Multiply $4$ by $1$ .

$90 = 4 + 4 e^{- x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $4 + 4 e^{- x} = 90$ .

$4 + 4 e^{- x} = 90$

Step 3.2

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

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Step 3.2.1

Subtract $4$ from both sides of the equation.

$4 e^{- x} = 90 - 4$

Step 3.2.2

Subtract $4$ from $90$ .

$4 e^{- x} = 86$

Step 3.3

Divide each term in $4 e^{- x} = 86$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 e^{- x} = 86$ by $4$ .

$\frac{4 e^{- x}}{4} = \frac{86}{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 e^{- x}}{4} = \frac{86}{4}$

Step 3.3.2.1.2

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

$e^{- x} = \frac{86}{4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $86$ and $4$ .

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Step 3.3.3.1.1

Factor $2$ out of $86$ .

$e^{- x} = \frac{2 (43)}{4}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $2$ out of $4$ .

$e^{- x} = \frac{2 \cdot 43}{2 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$e^{- x} = \frac{2 \cdot 43}{2 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$e^{- x} = \frac{43}{2}$

Step 3.4

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

$\ln (e^{- x}) = \ln (\frac{43}{2})$

Step 3.5

Expand the left side.

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Step 3.5.1

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

$- x \ln (e) = \ln (\frac{43}{2})$

Step 3.5.2

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

$- x \cdot 1 = \ln (\frac{43}{2})$

Step 3.5.3

Multiply $- 1$ by $1$ .

$- x = \ln (\frac{43}{2})$

Step 3.6

Divide each term in $- x = \ln (\frac{43}{2})$ by $- 1$ and simplify.

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Step 3.6.1

Divide each term in $- x = \ln (\frac{43}{2})$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (\frac{43}{2})}{- 1}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (\frac{43}{2})}{- 1}$

Step 3.6.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{43}{2})}{- 1}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative one from the denominator of $\frac{\ln (\frac{43}{2})}{- 1}$ .

$x = - 1 \cdot \ln (\frac{43}{2})$

Step 3.6.3.2

Rewrite $- 1 \cdot \ln (\frac{43}{2})$ as $- \ln (\frac{43}{2})$ .

$x = - \ln (\frac{43}{2})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \ln (\frac{43}{2})$

Decimal Form:

$x = - 3.06805293 \dots$