Finite Math Examples

$\frac{0.000001}{202} = \frac{e^{- 0.49028 t}}{202}$

Step 1

Rewrite the equation as $\frac{e^{- 0.49028 t}}{202} = \frac{0.000001}{202}$ .

$\frac{e^{- 0.49028 t}}{202} = \frac{0.000001}{202}$

Step 2

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

$e^{- 0.49028 t} = 0.000001$

Step 3

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

$\ln (e^{- 0.49028 t}) = \ln (0.000001)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{- 0.49028 t})$ by moving $- 0.49028 t$ outside the logarithm.

$- 0.49028 t \ln (e) = \ln (0.000001)$

Step 4.2

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

$- 0.49028 t \cdot 1 = \ln (0.000001)$

Step 4.3

Multiply $- 0.49028$ by $1$ .

$- 0.49028 t = \ln (0.000001)$

Step 5

Divide each term in $- 0.49028 t = \ln (0.000001)$ by $- 0.49028$ and simplify.

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Step 5.1

Divide each term in $- 0.49028 t = \ln (0.000001)$ by $- 0.49028$ .

$\frac{- 0.49028 t}{- 0.49028} = \frac{\ln (0.000001)}{- 0.49028}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.49028$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.49028 t}{- 0.49028} = \frac{\ln (0.000001)}{- 0.49028}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.000001)}{- 0.49028}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.000001)}{0.49028}$

Step 5.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.000001)}{0.49028}$

Step 5.3.3

Log base $2.71828182$ of $0.000001$ is approximately $- 13.81551055$ .

$t = - \frac{- 13.81551055}{0.49028}$

Step 5.3.4

Divide $- 13.81551055$ by $0.49028$ .

$t = - - 28.17881732$

Step 5.3.5

Multiply $- 1$ by $- 28.17881732$ .

$t = 28.17881732$