Finite Math Examples

$65 = \frac{75}{1 + 5 e^{- 0.8 x}}$

Step 1

Rewrite the equation as $\frac{75}{1 + 5 e^{- 0.8 x}} = 65$ .

$\frac{75}{1 + 5 e^{- 0.8 x}} = 65$

Step 2

Multiply both sides by $1 + 5 e^{- 0.8 x}$ .

$\frac{75}{1 + 5 e^{- 0.8 x}} (1 + 5 e^{- 0.8 x}) = 65 (1 + 5 e^{- 0.8 x})$

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 $1 + 5 e^{- 0.8 x}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{75}{1 + 5 e^{- 0.8 x}} (1 + 5 e^{- 0.8 x}) = 65 (1 + 5 e^{- 0.8 x})$

Step 3.1.1.2

Rewrite the expression.

$75 = 65 (1 + 5 e^{- 0.8 x})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $65 (1 + 5 e^{- 0.8 x})$ .

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Step 3.2.1.1

Apply the distributive property.

$75 = 65 \cdot 1 + 65 (5 e^{- 0.8 x})$

Step 3.2.1.2

Multiply.

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Step 3.2.1.2.1

Multiply $65$ by $1$ .

$75 = 65 + 65 (5 e^{- 0.8 x})$

Step 3.2.1.2.2

Multiply $5$ by $65$ .

$75 = 65 + 325 e^{- 0.8 x}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $65 + 325 e^{- 0.8 x} = 75$ .

$65 + 325 e^{- 0.8 x} = 75$

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 $65$ from both sides of the equation.

$325 e^{- 0.8 x} = 75 - 65$

Step 4.2.2

Subtract $65$ from $75$ .

$325 e^{- 0.8 x} = 10$

Step 4.3

Divide each term in $325 e^{- 0.8 x} = 10$ by $325$ and simplify.

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Step 4.3.1

Divide each term in $325 e^{- 0.8 x} = 10$ by $325$ .

$\frac{325 e^{- 0.8 x}}{325} = \frac{10}{325}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $325$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{325 e^{- 0.8 x}}{325} = \frac{10}{325}$

Step 4.3.2.1.2

Divide $e^{- 0.8 x}$ by $1$ .

$e^{- 0.8 x} = \frac{10}{325}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $10$ and $325$ .

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Step 4.3.3.1.1

Factor $5$ out of $10$ .

$e^{- 0.8 x} = \frac{5 (2)}{325}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $5$ out of $325$ .

$e^{- 0.8 x} = \frac{5 \cdot 2}{5 \cdot 65}$

Step 4.3.3.1.2.2

Cancel the common factor.

$e^{- 0.8 x} = \frac{5 \cdot 2}{5 \cdot 65}$

Step 4.3.3.1.2.3

Rewrite the expression.

$e^{- 0.8 x} = \frac{2}{65}$

Step 4.4

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

$\ln (e^{- 0.8 x}) = \ln (\frac{2}{65})$

Step 4.5

Expand the left side.

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Step 4.5.1

Expand $\ln (e^{- 0.8 x})$ by moving $- 0.8 x$ outside the logarithm.

$- 0.8 x \ln (e) = \ln (\frac{2}{65})$

Step 4.5.2

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

$- 0.8 x \cdot 1 = \ln (\frac{2}{65})$

Step 4.5.3

Multiply $- 0.8$ by $1$ .

$- 0.8 x = \ln (\frac{2}{65})$

Step 4.6

Divide each term in $- 0.8 x = \ln (\frac{2}{65})$ by $- 0.8$ and simplify.

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Step 4.6.1

Divide each term in $- 0.8 x = \ln (\frac{2}{65})$ by $- 0.8$ .

$\frac{- 0.8 x}{- 0.8} = \frac{\ln (\frac{2}{65})}{- 0.8}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $- 0.8$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{- 0.8 x}{- 0.8} = \frac{\ln (\frac{2}{65})}{- 0.8}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{2}{65})}{- 0.8}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{2}{65})}{0.8}$

Step 4.6.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (\frac{2}{65})}{0.8}$

Step 4.6.3.3

Divide $2$ by $65$ .

$x = - \frac{\log_{2.71828182} (0.0\overline{307692})}{0.8}$

Step 4.6.3.4

Log base $2.71828182$ of $0.0\overline{307692}$ is approximately $- 3.48124008$ .

$x = - \frac{- 3.48124008}{0.8}$

Step 4.6.3.5

Divide $- 3.48124008$ by $0.8$ .

$x = - - 4.35155011$

Step 4.6.3.6

Multiply $- 1$ by $- 4.35155011$ .

$x = 4.35155011$