Finite Math Examples

$85 = 100 - 100 e^{- 0.08 x}$

Step 1

Rewrite the equation as $100 - 100 e^{- 0.08 x} = 85$ .

$100 - 100 e^{- 0.08 x} = 85$

Step 2

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

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Step 2.1

Subtract $100$ from both sides of the equation.

$- 100 e^{- 0.08 x} = 85 - 100$

Step 2.2

Subtract $100$ from $85$ .

$- 100 e^{- 0.08 x} = - 15$

Step 3

Divide each term in $- 100 e^{- 0.08 x} = - 15$ by $- 100$ and simplify.

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Step 3.1

Divide each term in $- 100 e^{- 0.08 x} = - 15$ by $- 100$ .

$\frac{- 100 e^{- 0.08 x}}{- 100} = \frac{- 15}{- 100}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 100$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 100 e^{- 0.08 x}}{- 100} = \frac{- 15}{- 100}$

Step 3.2.1.2

Divide $e^{- 0.08 x}$ by $1$ .

$e^{- 0.08 x} = \frac{- 15}{- 100}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $- 15$ and $- 100$ .

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Step 3.3.1.1

Factor $- 5$ out of $- 15$ .

$e^{- 0.08 x} = \frac{- 5 (3)}{- 100}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $- 5$ out of $- 100$ .

$e^{- 0.08 x} = \frac{- 5 \cdot 3}{- 5 \cdot 20}$

Step 3.3.1.2.2

Cancel the common factor.

$e^{- 0.08 x} = \frac{- 5 \cdot 3}{- 5 \cdot 20}$

Step 3.3.1.2.3

Rewrite the expression.

$e^{- 0.08 x} = \frac{3}{20}$

Step 4

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

$\ln (e^{- 0.08 x}) = \ln (\frac{3}{20})$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{- 0.08 x})$ by moving $- 0.08 x$ outside the logarithm.

$- 0.08 x \ln (e) = \ln (\frac{3}{20})$

Step 5.2

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

$- 0.08 x \cdot 1 = \ln (\frac{3}{20})$

Step 5.3

Multiply $- 0.08$ by $1$ .

$- 0.08 x = \ln (\frac{3}{20})$

Step 6

Divide each term in $- 0.08 x = \ln (\frac{3}{20})$ by $- 0.08$ and simplify.

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Step 6.1

Divide each term in $- 0.08 x = \ln (\frac{3}{20})$ by $- 0.08$ .

$\frac{- 0.08 x}{- 0.08} = \frac{\ln (\frac{3}{20})}{- 0.08}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 0.08$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 0.08 x}{- 0.08} = \frac{\ln (\frac{3}{20})}{- 0.08}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{3}{20})}{- 0.08}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{3}{20})}{0.08}$

Step 6.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (\frac{3}{20})}{0.08}$

Step 6.3.3

Divide $3$ by $20$ .

$x = - \frac{\log_{2.71828182} (0.15)}{0.08}$

Step 6.3.4

Log base $2.71828182$ of $0.15$ is approximately $- 1.89711998$ .

$x = - \frac{- 1.89711998}{0.08}$

Step 6.3.5

Divide $- 1.89711998$ by $0.08$ .

$x = - - 23.71399981$

Step 6.3.6

Multiply $- 1$ by $- 23.71399981$ .

$x = 23.71399981$