Basic Math Examples

$\frac{5}{10} = \frac{10 (1 - e^{- 0.2 t})}{10}$

Step 1

Rewrite the equation as $\frac{10 (1 - e^{- 0.2 t})}{10} = \frac{5}{10}$ .

$\frac{10 (1 - e^{- 0.2 t})}{10} = \frac{5}{10}$

Step 2

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

$10 (1 - e^{- 0.2 t}) = 5$

Step 3

Divide each term in $10 (1 - e^{- 0.2 t}) = 5$ by $10$ and simplify.

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Step 3.1

Divide each term in $10 (1 - e^{- 0.2 t}) = 5$ by $10$ .

$\frac{10 (1 - e^{- 0.2 t})}{10} = \frac{5}{10}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $10$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{10 (1 - e^{- 0.2 t})}{10} = \frac{5}{10}$

Step 3.2.1.2

Divide $1 - e^{- 0.2 t}$ by $1$ .

$1 - e^{- 0.2 t} = \frac{5}{10}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $5$ and $10$ .

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Step 3.3.1.1

Factor $5$ out of $5$ .

$1 - e^{- 0.2 t} = \frac{5 (1)}{10}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $5$ out of $10$ .

$1 - e^{- 0.2 t} = \frac{5 \cdot 1}{5 \cdot 2}$

Step 3.3.1.2.2

Cancel the common factor.

$1 - e^{- 0.2 t} = \frac{5 \cdot 1}{5 \cdot 2}$

Step 3.3.1.2.3

Rewrite the expression.

$1 - e^{- 0.2 t} = \frac{1}{2}$

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the equation.

$- e^{- 0.2 t} = \frac{1}{2} - 1$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- e^{- 0.2 t} = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 4.3

Combine $- 1$ and $\frac{2}{2}$ .

$- e^{- 0.2 t} = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 4.4

Combine the numerators over the common denominator.

$- e^{- 0.2 t} = \frac{1 - 1 \cdot 2}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$- e^{- 0.2 t} = \frac{1 - 2}{2}$

Step 4.5.2

Subtract $2$ from $1$ .

$- e^{- 0.2 t} = \frac{- 1}{2}$

Step 4.6

Move the negative in front of the fraction.

$- e^{- 0.2 t} = - \frac{1}{2}$

Step 5

Divide each term in $- e^{- 0.2 t} = - \frac{1}{2}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- e^{- 0.2 t} = - \frac{1}{2}$ by $- 1$ .

$\frac{- e^{- 0.2 t}}{- 1} = \frac{- \frac{1}{2}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.2 t}}{1} = \frac{- \frac{1}{2}}{- 1}$

Step 5.2.2

Divide $e^{- 0.2 t}$ by $1$ .

$e^{- 0.2 t} = \frac{- \frac{1}{2}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$e^{- 0.2 t} = \frac{\frac{1}{2}}{1}$

Step 5.3.2

Divide $\frac{1}{2}$ by $1$ .

$e^{- 0.2 t} = \frac{1}{2}$

Step 6

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

$\ln (e^{- 0.2 t}) = \ln (\frac{1}{2})$

Step 7

Expand the left side.

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Step 7.1

Expand $\ln (e^{- 0.2 t})$ by moving $- 0.2 t$ outside the logarithm.

$- 0.2 t \ln (e) = \ln (\frac{1}{2})$

Step 7.2

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

$- 0.2 t \cdot 1 = \ln (\frac{1}{2})$

Step 7.3

Multiply $- 0.2$ by $1$ .

$- 0.2 t = \ln (\frac{1}{2})$

Step 8

Divide each term in $- 0.2 t = \ln (\frac{1}{2})$ by $- 0.2$ and simplify.

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Step 8.1

Divide each term in $- 0.2 t = \ln (\frac{1}{2})$ by $- 0.2$ .

$\frac{- 0.2 t}{- 0.2} = \frac{\ln (\frac{1}{2})}{- 0.2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 0.2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 0.2 t}{- 0.2} = \frac{\ln (\frac{1}{2})}{- 0.2}$

Step 8.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1}{2})}{- 0.2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{1}{2})}{0.2}$

Step 8.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{1}{2})}{0.2}$

Step 8.3.3

Divide $1$ by $2$ .

$t = - \frac{\log_{2.71828182} (0.5)}{0.2}$

Step 8.3.4

Log base $2.71828182$ of $0.5$ is approximately $- 0.69314718$ .

$t = - \frac{- 0.69314718}{0.2}$

Step 8.3.5

Divide $- 0.69314718$ by $0.2$ .

$t = - - 3.4657359$

Step 8.3.6

Multiply $- 1$ by $- 3.4657359$ .

$t = 3.4657359$