Precalculus Examples

$0.2 = 1 - e^{(- 0.08) (t)}$

Step 1

Rewrite the equation as $1 - e^{(- 0.08) (t)} = 0.2$ .

$1 - e^{(- 0.08) (t)} = 0.2$

Step 2

Multiply $- 0.08$ by $t$ .

$1 - e^{- 0.08 t} = 0.2$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$- e^{- 0.08 t} = 0.2 - 1$

Step 3.2

Subtract $1$ from $0.2$ .

$- e^{- 0.08 t} = - 0.8$

Step 4

Divide each term in $- e^{- 0.08 t} = - 0.8$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{- 0.08 t} = - 0.8$ by $- 1$ .

$\frac{- e^{- 0.08 t}}{- 1} = \frac{- 0.8}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.08 t}}{1} = \frac{- 0.8}{- 1}$

Step 4.2.2

Divide $e^{- 0.08 t}$ by $1$ .

$e^{- 0.08 t} = \frac{- 0.8}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 0.8$ by $- 1$ .

$e^{- 0.08 t} = 0.8$

Step 5

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

$\ln (e^{- 0.08 t}) = \ln (0.8)$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.08 t})$ by moving $- 0.08 t$ outside the logarithm.

$- 0.08 t \ln (e) = \ln (0.8)$

Step 6.2

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

$- 0.08 t \cdot 1 = \ln (0.8)$

Step 6.3

Multiply $- 0.08$ by $1$ .

$- 0.08 t = \ln (0.8)$

Step 7

Divide each term in $- 0.08 t = \ln (0.8)$ by $- 0.08$ and simplify.

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Step 7.1

Divide each term in $- 0.08 t = \ln (0.8)$ by $- 0.08$ .

$\frac{- 0.08 t}{- 0.08} = \frac{\ln (0.8)}{- 0.08}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.08$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.08 t}{- 0.08} = \frac{\ln (0.8)}{- 0.08}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.8)}{- 0.08}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.8)}{0.08}$

Step 7.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.8)}{0.08}$

Step 7.3.3

Log base $2.71828182$ of $0.8$ is approximately $- 0.22314355$ .

$t = - \frac{- 0.22314355}{0.08}$

Step 7.3.4

Divide $- 0.22314355$ by $0.08$ .

$t = - - 2.78929439$

Step 7.3.5

Multiply $- 1$ by $- 2.78929439$ .

$t = 2.78929439$