Finite Math Examples

$\frac{200}{1 + 8 e^{- 0.3 t}} = 100$

Step 1

Multiply both sides by $1 + 8 e^{- 0.3 t}$ .

$\frac{200}{1 + 8 e^{- 0.3 t}} (1 + 8 e^{- 0.3 t}) = 100 (1 + 8 e^{- 0.3 t})$

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 $1 + 8 e^{- 0.3 t}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{200}{1 + 8 e^{- 0.3 t}} (1 + 8 e^{- 0.3 t}) = 100 (1 + 8 e^{- 0.3 t})$

Step 2.1.1.2

Rewrite the expression.

$200 = 100 (1 + 8 e^{- 0.3 t})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $100 (1 + 8 e^{- 0.3 t})$ .

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Step 2.2.1.1

Apply the distributive property.

$200 = 100 \cdot 1 + 100 (8 e^{- 0.3 t})$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $100$ by $1$ .

$200 = 100 + 100 (8 e^{- 0.3 t})$

Step 2.2.1.2.2

Multiply $8$ by $100$ .

$200 = 100 + 800 e^{- 0.3 t}$

Step 3

Solve for $t$ .

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Step 3.1

Rewrite the equation as $100 + 800 e^{- 0.3 t} = 200$ .

$100 + 800 e^{- 0.3 t} = 200$

Step 3.2

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

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Step 3.2.1

Subtract $100$ from both sides of the equation.

$800 e^{- 0.3 t} = 200 - 100$

Step 3.2.2

Subtract $100$ from $200$ .

$800 e^{- 0.3 t} = 100$

Step 3.3

Divide each term in $800 e^{- 0.3 t} = 100$ by $800$ and simplify.

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Step 3.3.1

Divide each term in $800 e^{- 0.3 t} = 100$ by $800$ .

$\frac{800 e^{- 0.3 t}}{800} = \frac{100}{800}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $800$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{800 e^{- 0.3 t}}{800} = \frac{100}{800}$

Step 3.3.2.1.2

Divide $e^{- 0.3 t}$ by $1$ .

$e^{- 0.3 t} = \frac{100}{800}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $100$ and $800$ .

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Step 3.3.3.1.1

Factor $100$ out of $100$ .

$e^{- 0.3 t} = \frac{100 (1)}{800}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $100$ out of $800$ .

$e^{- 0.3 t} = \frac{100 \cdot 1}{100 \cdot 8}$

Step 3.3.3.1.2.2

Cancel the common factor.

$e^{- 0.3 t} = \frac{100 \cdot 1}{100 \cdot 8}$

Step 3.3.3.1.2.3

Rewrite the expression.

$e^{- 0.3 t} = \frac{1}{8}$

Step 3.4

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

$\ln (e^{- 0.3 t}) = \ln (\frac{1}{8})$

Step 3.5

Expand the left side.

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Step 3.5.1

Expand $\ln (e^{- 0.3 t})$ by moving $- 0.3 t$ outside the logarithm.

$- 0.3 t \ln (e) = \ln (\frac{1}{8})$

Step 3.5.2

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

$- 0.3 t \cdot 1 = \ln (\frac{1}{8})$

Step 3.5.3

Multiply $- 0.3$ by $1$ .

$- 0.3 t = \ln (\frac{1}{8})$

Step 3.6

Divide each term in $- 0.3 t = \ln (\frac{1}{8})$ by $- 0.3$ and simplify.

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Step 3.6.1

Divide each term in $- 0.3 t = \ln (\frac{1}{8})$ by $- 0.3$ .

$\frac{- 0.3 t}{- 0.3} = \frac{\ln (\frac{1}{8})}{- 0.3}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $- 0.3$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{- 0.3 t}{- 0.3} = \frac{\ln (\frac{1}{8})}{- 0.3}$

Step 3.6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1}{8})}{- 0.3}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{1}{8})}{0.3}$

Step 3.6.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{1}{8})}{0.3}$

Step 3.6.3.3

Divide $1$ by $8$ .

$t = - \frac{\log_{2.71828182} (0.125)}{0.3}$

Step 3.6.3.4

Log base $2.71828182$ of $0.125$ is approximately $- 2.07944154$ .

$t = - \frac{- 2.07944154}{0.3}$

Step 3.6.3.5

Divide $- 2.07944154$ by $0.3$ .

$t = - - 6.9314718$

Step 3.6.3.6

Multiply $- 1$ by $- 6.9314718$ .

$t = 6.9314718$