Finite Math Examples

$0.64 = 1 - 0.967 e^{- 0.17 t}$

Step 1

Rewrite the equation as $1 - 0.967 e^{- 0.17 t} = 0.64$ .

$1 - 0.967 e^{- 0.17 t} = 0.64$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

$- 0.967 e^{- 0.17 t} = 0.64 - 1$

Step 2.2

Subtract $1$ from $0.64$ .

$- 0.967 e^{- 0.17 t} = - 0.36$

Step 3

Divide each term in $- 0.967 e^{- 0.17 t} = - 0.36$ by $- 0.967$ and simplify.

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Step 3.1

Divide each term in $- 0.967 e^{- 0.17 t} = - 0.36$ by $- 0.967$ .

$\frac{- 0.967 e^{- 0.17 t}}{- 0.967} = \frac{- 0.36}{- 0.967}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 0.967$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 0.967 e^{- 0.17 t}}{- 0.967} = \frac{- 0.36}{- 0.967}$

Step 3.2.1.2

Divide $e^{- 0.17 t}$ by $1$ .

$e^{- 0.17 t} = \frac{- 0.36}{- 0.967}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 0.36$ by $- 0.967$ .

$e^{- 0.17 t} = 0.37228541$

Step 4

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

$\ln (e^{- 0.17 t}) = \ln (0.37228541)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{- 0.17 t})$ by moving $- 0.17 t$ outside the logarithm.

$- 0.17 t \ln (e) = \ln (0.37228541)$

Step 5.2

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

$- 0.17 t \cdot 1 = \ln (0.37228541)$

Step 5.3

Multiply $- 0.17$ by $1$ .

$- 0.17 t = \ln (0.37228541)$

Step 6

Divide each term in $- 0.17 t = \ln (0.37228541)$ by $- 0.17$ and simplify.

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Step 6.1

Divide each term in $- 0.17 t = \ln (0.37228541)$ by $- 0.17$ .

$\frac{- 0.17 t}{- 0.17} = \frac{\ln (0.37228541)}{- 0.17}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 0.17$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 0.17 t}{- 0.17} = \frac{\ln (0.37228541)}{- 0.17}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.37228541)}{- 0.17}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.37228541)}{0.17}$

Step 6.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.37228541)}{0.17}$

Step 6.3.3

Log base $2.71828182$ of $0.37228541$ is approximately $- 0.98809446$ .

$t = - \frac{- 0.98809446}{0.17}$

Step 6.3.4

Divide $- 0.98809446$ by $0.17$ .

$t = - - 5.81232037$

Step 6.3.5

Multiply $- 1$ by $- 5.81232037$ .

$t = 5.81232037$