Finite Math Examples

$0.03 = 0.14 (1 - e^{0.18 (t)})$

Step 1

Rewrite the equation as $0.14 (1 - e^{0.18 t}) = 0.03$ .

$0.14 (1 - e^{0.18 t}) = 0.03$

Step 2

Divide each term in $0.14 (1 - e^{0.18 t}) = 0.03$ by $0.14$ and simplify.

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Step 2.1

Divide each term in $0.14 (1 - e^{0.18 t}) = 0.03$ by $0.14$ .

$\frac{0.14 (1 - e^{0.18 t})}{0.14} = \frac{0.03}{0.14}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.14$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.14 (1 - e^{0.18 t})}{0.14} = \frac{0.03}{0.14}$

Step 2.2.1.2

Divide $1 - e^{0.18 t}$ by $1$ .

$1 - e^{0.18 t} = \frac{0.03}{0.14}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0.03$ by $0.14$ .

$1 - e^{0.18 t} = 0.2\overline{142857}$

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.

$- e^{0.18 t} = 0.2\overline{142857} - 1$

Step 3.2

Subtract $1$ from $0.2\overline{142857}$ .

$- e^{0.18 t} = - 0.7\overline{857142}$

Step 4

Divide each term in $- e^{0.18 t} = - 0.7\overline{857142}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{0.18 t} = - 0.7\overline{857142}$ by $- 1$ .

$\frac{- e^{0.18 t}}{- 1} = \frac{- 0.7\overline{857142}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{0.18 t}}{1} = \frac{- 0.7\overline{857142}}{- 1}$

Step 4.2.2

Divide $e^{0.18 t}$ by $1$ .

$e^{0.18 t} = \frac{- 0.7\overline{857142}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 0.7\overline{857142}$ by $- 1$ .

$e^{0.18 t} = 0.7\overline{857142}$

Step 5

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

$\ln (e^{0.18 t}) = \ln (0.7\overline{857142})$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{0.18 t})$ by moving $0.18 t$ outside the logarithm.

$0.18 t \ln (e) = \ln (0.7\overline{857142})$

Step 6.2

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

$0.18 t \cdot 1 = \ln (0.7\overline{857142})$

Step 6.3

Multiply $0.18$ by $1$ .

$0.18 t = \ln (0.7\overline{857142})$

Step 7

Divide each term in $0.18 t = \ln (0.7\overline{857142})$ by $0.18$ and simplify.

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Step 7.1

Divide each term in $0.18 t = \ln (0.7\overline{857142})$ by $0.18$ .

$\frac{0.18 t}{0.18} = \frac{\ln (0.7\overline{857142})}{0.18}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $0.18$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{0.18 t}{0.18} = \frac{\ln (0.7\overline{857142})}{0.18}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.7\overline{857142})}{0.18}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (0.7\overline{857142})}{0.18}$

Step 7.3.2

Log base $2.71828182$ of $0.7\overline{857142}$ is approximately $- 0.24116205$ .

$t = \frac{- 0.24116205}{0.18}$

Step 7.3.3

Divide $- 0.24116205$ by $0.18$ .

$t = - 1.3397892$