Precalculus Examples

$\frac{400}{1 + e^{- x}} = 350$

Step 1

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

$\frac{400}{1 + e^{- x}} (1 + e^{- x}) = 350 (1 + e^{- x})$

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.

$\frac{400}{1 + e^{- x}} (1 + e^{- x}) = 350 (1 + e^{- x})$

Step 2.1.1.2

Rewrite the expression.

$400 = 350 (1 + e^{- x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $350 (1 + e^{- x})$ .

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Step 2.2.1.1

Apply the distributive property.

$400 = 350 \cdot 1 + 350 e^{- x}$

Step 2.2.1.2

Multiply $350$ by $1$ .

$400 = 350 + 350 e^{- x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $350 + 350 e^{- x} = 400$ .

$350 + 350 e^{- x} = 400$

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 $350$ from both sides of the equation.

$350 e^{- x} = 400 - 350$

Step 3.2.2

Subtract $350$ from $400$ .

$350 e^{- x} = 50$

Step 3.3

Divide each term in $350 e^{- x} = 50$ by $350$ and simplify.

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Step 3.3.1

Divide each term in $350 e^{- x} = 50$ by $350$ .

$\frac{350 e^{- x}}{350} = \frac{50}{350}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $350$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{350 e^{- x}}{350} = \frac{50}{350}$

Step 3.3.2.1.2

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

$e^{- x} = \frac{50}{350}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $50$ and $350$ .

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Step 3.3.3.1.1

Factor $50$ out of $50$ .

$e^{- x} = \frac{50 (1)}{350}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $50$ out of $350$ .

$e^{- x} = \frac{50 \cdot 1}{50 \cdot 7}$

Step 3.3.3.1.2.2

Cancel the common factor.

$e^{- x} = \frac{50 \cdot 1}{50 \cdot 7}$

Step 3.3.3.1.2.3

Rewrite the expression.

$e^{- x} = \frac{1}{7}$

Step 3.4

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

$\ln (e^{- x}) = \ln (\frac{1}{7})$

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.

$- x \ln (e) = \ln (\frac{1}{7})$

Step 3.5.2

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

$- x \cdot 1 = \ln (\frac{1}{7})$

Step 3.5.3

Multiply $- 1$ by $1$ .

$- x = \ln (\frac{1}{7})$

Step 3.6

Divide each term in $- x = \ln (\frac{1}{7})$ by $- 1$ and simplify.

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Step 3.6.1

Divide each term in $- x = \ln (\frac{1}{7})$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (\frac{1}{7})}{- 1}$

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.

$\frac{x}{1} = \frac{\ln (\frac{1}{7})}{- 1}$

Step 3.6.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{7})}{- 1}$

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 $\frac{\ln (\frac{1}{7})}{- 1}$ .

$x = - 1 \cdot \ln (\frac{1}{7})$

Step 3.6.3.2

Rewrite $- 1 \cdot \ln (\frac{1}{7})$ as $- \ln (\frac{1}{7})$ .

$x = - \ln (\frac{1}{7})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \ln (\frac{1}{7})$

Decimal Form:

$x = 1.94591014 \dots$