Calculus Examples

$10 {(1 + e^{- x})}^{- 1} = 3$

Step 1

Divide each term in $10 {(1 + e^{- x})}^{- 1} = 3$ by $10$ and simplify.

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Step 1.1

Divide each term in $10 {(1 + e^{- x})}^{- 1} = 3$ by $10$ .

$\frac{10 {(1 + e^{- x})}^{- 1}}{10} = \frac{3}{10}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Move ${(1 + e^{- x})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10}{10 (1 + e^{- x})} = \frac{3}{10}$

Step 1.2.2

Cancel the common factor of $10$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{10}{10 (1 + e^{- x})} = \frac{3}{10}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{1 + e^{- x}} = \frac{3}{10}$

Step 2

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

$\frac{1}{1 + e^{- x}} (1 + e^{- x}) = \frac{3}{10} (1 + e^{- x})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$\frac{1}{1 + e^{- x}} (1 + e^{- x}) = \frac{3}{10} (1 + e^{- x})$

Step 3.1.1.2

Rewrite the expression.

$1 = \frac{3}{10} (1 + e^{- x})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{3}{10} (1 + e^{- x})$ .

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Step 3.2.1.1

Apply the distributive property.

$1 = \frac{3}{10} \cdot 1 + \frac{3}{10} e^{- x}$

Step 3.2.1.2

Multiply $\frac{3}{10}$ by $1$ .

$1 = \frac{3}{10} + \frac{3}{10} e^{- x}$

Step 3.2.1.3

Combine $\frac{3}{10}$ and $e^{- x}$ .

$1 = \frac{3}{10} + \frac{3 e^{- x}}{10}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{3}{10} + \frac{3 e^{- x}}{10} = 1$ .

$\frac{3}{10} + \frac{3 e^{- x}}{10} = 1$

Step 4.2

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

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Step 4.2.1

Subtract $\frac{3}{10}$ from both sides of the equation.

$\frac{3 e^{- x}}{10} = 1 - \frac{3}{10}$

Step 4.2.2

Write $1$ as a fraction with a common denominator.

$\frac{3 e^{- x}}{10} = \frac{10}{10} - \frac{3}{10}$

Step 4.2.3

Combine the numerators over the common denominator.

$\frac{3 e^{- x}}{10} = \frac{10 - 3}{10}$

Step 4.2.4

Subtract $3$ from $10$ .

$\frac{3 e^{- x}}{10} = \frac{7}{10}$

Step 4.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 e^{- x} = 7$

Step 4.4

Divide each term in $3 e^{- x} = 7$ by $3$ and simplify.

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Step 4.4.1

Divide each term in $3 e^{- x} = 7$ by $3$ .

$\frac{3 e^{- x}}{3} = \frac{7}{3}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{3 e^{- x}}{3} = \frac{7}{3}$

Step 4.4.2.1.2

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

$e^{- x} = \frac{7}{3}$

Step 4.5

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

$\ln (e^{- x}) = \ln (\frac{7}{3})$

Step 4.6

Expand the left side.

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Step 4.6.1

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

$- x \ln (e) = \ln (\frac{7}{3})$

Step 4.6.2

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

$- x \cdot 1 = \ln (\frac{7}{3})$

Step 4.6.3

Multiply $- 1$ by $1$ .

$- x = \ln (\frac{7}{3})$

Step 4.7

Divide each term in $- x = \ln (\frac{7}{3})$ by $- 1$ and simplify.

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Step 4.7.1

Divide each term in $- x = \ln (\frac{7}{3})$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (\frac{7}{3})}{- 1}$

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (\frac{7}{3})}{- 1}$

Step 4.7.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{7}{3})}{- 1}$

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Move the negative one from the denominator of $\frac{\ln (\frac{7}{3})}{- 1}$ .

$x = - 1 \cdot \ln (\frac{7}{3})$

Step 4.7.3.2

Rewrite $- 1 \cdot \ln (\frac{7}{3})$ as $- \ln (\frac{7}{3})$ .

$x = - \ln (\frac{7}{3})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \ln (\frac{7}{3})$

Decimal Form:

$x = - 0.84729786 \dots$