Calculus Examples

$\frac{1}{1 + a e^{- k t}} = \frac{1}{2}$

Step 1

Multiply both sides by $1 + a e^{- k t}$ .

$\frac{1}{1 + a e^{- k t}} (1 + a e^{- k t}) = \frac{1}{2} (1 + a e^{- k t})$

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 $1 + a e^{- k t}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{1}{1 + a e^{- k t}} (1 + a e^{- k t}) = \frac{1}{2} (1 + a e^{- k t})$

Step 2.1.1.2

Rewrite the expression.

$1 = \frac{1}{2} (1 + a e^{- k t})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1}{2} (1 + a e^{- k t})$ .

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Step 2.2.1.1

Apply the distributive property.

$1 = \frac{1}{2} \cdot 1 + \frac{1}{2} (a e^{- k t})$

Step 2.2.1.2

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

$1 = \frac{1}{2} + \frac{1}{2} (a e^{- k t})$

Step 2.2.1.3

Multiply $\frac{1}{2} (a e^{- k t})$ .

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Step 2.2.1.3.1

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

$1 = \frac{1}{2} + \frac{a}{2} e^{- k t}$

Step 2.2.1.3.2

Combine $\frac{a}{2}$ and $e^{- k t}$ .

$1 = \frac{1}{2} + \frac{a e^{- k t}}{2}$

Step 2.2.1.4

Reorder $\frac{1}{2}$ and $\frac{a e^{- k t}}{2}$ .

$1 = \frac{a e^{- k t}}{2} + \frac{1}{2}$

Step 3

Solve for $t$ .

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Step 3.1

Rewrite the equation as $\frac{a e^{- k t}}{2} + \frac{1}{2} = 1$ .

$\frac{a e^{- k t}}{2} + \frac{1}{2} = 1$

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 $\frac{1}{2}$ from both sides of the equation.

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

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Subtract $1$ from $2$ .

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

Step 3.3

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

$a e^{- k t} = 1$

Step 3.4

Divide each term in $a e^{- k t} = 1$ by $a$ and simplify.

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Step 3.4.1

Divide each term in $a e^{- k t} = 1$ by $a$ .

$\frac{a e^{- k t}}{a} = \frac{1}{a}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $a$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{a e^{- k t}}{a} = \frac{1}{a}$

Step 3.4.2.1.2

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

$e^{- k t} = \frac{1}{a}$

Step 3.5

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

$\ln (e^{- k t}) = \ln (\frac{1}{a})$

Step 3.6

Expand the left side.

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Step 3.6.1

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

$- k t \ln (e) = \ln (\frac{1}{a})$

Step 3.6.2

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

$- k t \cdot 1 = \ln (\frac{1}{a})$

Step 3.6.3

Multiply $- 1$ by $1$ .

$- k t = \ln (\frac{1}{a})$

Step 3.7

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

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Step 3.7.1

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

$\frac{- k t}{- k} = \frac{\ln (\frac{1}{a})}{- k}$

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Dividing two negative values results in a positive value.

$\frac{k t}{k} = \frac{\ln (\frac{1}{a})}{- k}$

Step 3.7.2.2

Cancel the common factor of $k$ .

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Step 3.7.2.2.1

Cancel the common factor.

$\frac{k t}{k} = \frac{\ln (\frac{1}{a})}{- k}$

Step 3.7.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1}{a})}{- k}$

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{1}{a})}{k}$