Finite Math Examples

$82 = 100 - 100 e^{- 0.04 x}$

Step 1

Rewrite the equation as $100 - 100 e^{- 0.04 x} = 82$ .

$100 - 100 e^{- 0.04 x} = 82$

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.

$- 100 e^{- 0.04 x} = 82 - 100$

Step 2.2

Subtract $100$ from $82$ .

$- 100 e^{- 0.04 x} = - 18$

Step 3

Divide each term in $- 100 e^{- 0.04 x} = - 18$ by $- 100$ and simplify.

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Step 3.1

Divide each term in $- 100 e^{- 0.04 x} = - 18$ by $- 100$ .

$\frac{- 100 e^{- 0.04 x}}{- 100} = \frac{- 18}{- 100}$

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.

$\frac{- 100 e^{- 0.04 x}}{- 100} = \frac{- 18}{- 100}$

Step 3.2.1.2

Divide $e^{- 0.04 x}$ by $1$ .

$e^{- 0.04 x} = \frac{- 18}{- 100}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $- 18$ and $- 100$ .

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Step 3.3.1.1

Factor $- 2$ out of $- 18$ .

$e^{- 0.04 x} = \frac{- 2 (9)}{- 100}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $- 2$ out of $- 100$ .

$e^{- 0.04 x} = \frac{- 2 \cdot 9}{- 2 \cdot 50}$

Step 3.3.1.2.2

Cancel the common factor.

$e^{- 0.04 x} = \frac{- 2 \cdot 9}{- 2 \cdot 50}$

Step 3.3.1.2.3

Rewrite the expression.

$e^{- 0.04 x} = \frac{9}{50}$

Step 4

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

$\ln (e^{- 0.04 x}) = \ln (\frac{9}{50})$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{- 0.04 x})$ by moving $- 0.04 x$ outside the logarithm.

$- 0.04 x \ln (e) = \ln (\frac{9}{50})$

Step 5.2

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

$- 0.04 x \cdot 1 = \ln (\frac{9}{50})$

Step 5.3

Multiply $- 0.04$ by $1$ .

$- 0.04 x = \ln (\frac{9}{50})$

Step 6

Divide each term in $- 0.04 x = \ln (\frac{9}{50})$ by $- 0.04$ and simplify.

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Step 6.1

Divide each term in $- 0.04 x = \ln (\frac{9}{50})$ by $- 0.04$ .

$\frac{- 0.04 x}{- 0.04} = \frac{\ln (\frac{9}{50})}{- 0.04}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 0.04$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 0.04 x}{- 0.04} = \frac{\ln (\frac{9}{50})}{- 0.04}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{9}{50})}{- 0.04}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{9}{50})}{0.04}$

Step 6.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (\frac{9}{50})}{0.04}$

Step 6.3.3

Divide $9$ by $50$ .

$x = - \frac{\log_{2.71828182} (0.18)}{0.04}$

Step 6.3.4

Log base $2.71828182$ of $0.18$ is approximately $- 1.71479842$ .

$x = - \frac{- 1.71479842}{0.04}$

Step 6.3.5

Divide $- 1.71479842$ by $0.04$ .

$x = - - 42.8699607$

Step 6.3.6

Multiply $- 1$ by $- 42.8699607$ .

$x = 42.8699607$