Calculus Examples

$N = 48 (1 - e^{- 0.07 t})$

Step 1

Rewrite the equation as $48 (1 - e^{- 0.07 t}) = N$ .

$48 (1 - e^{- 0.07 t}) = N$

Step 2

Divide each term in $48 (1 - e^{- 0.07 t}) = N$ by $48$ and simplify.

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Step 2.1

Divide each term in $48 (1 - e^{- 0.07 t}) = N$ by $48$ .

$\frac{48 (1 - e^{- 0.07 t})}{48} = \frac{N}{48}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $48$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{48 (1 - e^{- 0.07 t})}{48} = \frac{N}{48}$

Step 2.2.1.2

Divide $1 - e^{- 0.07 t}$ by $1$ .

$1 - e^{- 0.07 t} = \frac{N}{48}$

Step 3

Subtract $1$ from both sides of the equation.

$- e^{- 0.07 t} = \frac{N}{48} - 1$

Step 4

Divide each term in $- e^{- 0.07 t} = \frac{N}{48} - 1$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{- 0.07 t} = \frac{N}{48} - 1$ by $- 1$ .

$\frac{- e^{- 0.07 t}}{- 1} = \frac{\frac{N}{48}}{- 1} + \frac{- 1}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.07 t}}{1} = \frac{\frac{N}{48}}{- 1} + \frac{- 1}{- 1}$

Step 4.2.2

Divide $e^{- 0.07 t}$ by $1$ .

$e^{- 0.07 t} = \frac{\frac{N}{48}}{- 1} + \frac{- 1}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\frac{N}{48}}{- 1}$ .

$e^{- 0.07 t} = - 1 \cdot \frac{N}{48} + \frac{- 1}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \frac{N}{48}$ as $- \frac{N}{48}$ .

$e^{- 0.07 t} = - \frac{N}{48} + \frac{- 1}{- 1}$

Step 4.3.1.3

Divide $- 1$ by $- 1$ .

$e^{- 0.07 t} = - \frac{N}{48} + 1$

Step 5

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

$\ln (e^{- 0.07 t}) = \ln (- \frac{N}{48} + 1)$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.07 t})$ by moving $- 0.07 t$ outside the logarithm.

$- 0.07 t \ln (e) = \ln (- \frac{N}{48} + 1)$

Step 6.2

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

$- 0.07 t \cdot 1 = \ln (- \frac{N}{48} + 1)$

Step 6.3

Multiply $- 0.07$ by $1$ .

$- 0.07 t = \ln (- \frac{N}{48} + 1)$

Step 7

Divide each term in $- 0.07 t = \ln (- \frac{N}{48} + 1)$ by $- 0.07$ and simplify.

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Step 7.1

Divide each term in $- 0.07 t = \ln (- \frac{N}{48} + 1)$ by $- 0.07$ .

$\frac{- 0.07 t}{- 0.07} = \frac{\ln (- \frac{N}{48} + 1)}{- 0.07}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.07$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.07 t}{- 0.07} = \frac{\ln (- \frac{N}{48} + 1)}{- 0.07}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (- \frac{N}{48} + 1)}{- 0.07}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (- \frac{N}{48} + 1)}{0.07}$

Step 7.3.2

Multiply by $1$ .

$t = - \frac{1 \ln (- \frac{N}{48} + 1)}{0.07}$

Step 7.3.3

Factor $0.07$ out of $0.07$ .

$t = - \frac{1 \ln (- \frac{N}{48} + 1)}{0.07 (1)}$

Step 7.3.4

Separate fractions.

$t = - (\frac{1}{0.07} \cdot \frac{\ln (- \frac{N}{48} + 1)}{1})$

Step 7.3.5

Divide $1$ by $0.07$ .

$t = - (14.\overline{285714} \frac{\ln (- \frac{N}{48} + 1)}{1})$

Step 7.3.6

Divide $\ln (- \frac{N}{48} + 1)$ by $1$ .

$t = - (14.\overline{285714} \ln (- \frac{N}{48} + 1))$

Step 7.3.7

Multiply $14.\overline{285714}$ by $- 1$ .

$t = - 14.\overline{285714} \ln (- \frac{N}{48} + 1)$