Finite Math Examples

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

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

Step 1

Rewrite the equation as

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

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

\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

$e^{- 0.49028 t} = 0.000001$

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)

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

Step 4

Expand the left side.

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Step 4.1

Expand

\ln (e^{- 0.49028 t})

$\ln (e^{- 0.49028 t})$ by moving

- 0.49028 t

$- 0.49028 t$ outside the logarithm.

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

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

Step 4.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 4.3

Multiply

- 0.49028

$- 0.49028$ by

1

$1$ .

- 0.49028 t = \ln (0.000001)

$- 0.49028 t = \ln (0.000001)$

- 0.49028 t = \ln (0.000001)

$- 0.49028 t = \ln (0.000001)$

Step 5

Divide each term in

- 0.49028 t = \ln (0.000001)

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

- 0.49028

$- 0.49028$ and simplify.

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Step 5.1

Divide each term in

- 0.49028 t = \ln (0.000001)

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

- 0.49028

$- 0.49028$ .

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

$- 0.49028$ .

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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}

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

Step 5.2.1.2

Divide

t

$t$ by

1

$1$ .

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

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

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

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

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

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

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}

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

Step 5.3.2

Replace

e

$e$ with an approximation.

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

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

Step 5.3.3

Log base

2.71828182

$2.71828182$ of

0.000001

$0.000001$ is approximately

- 13.81551055

$- 13.81551055$ .

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

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

Step 5.3.4

Divide

- 13.81551055

$- 13.81551055$ by

0.49028

$0.49028$ .

t = - - 28.17881732

$t = - - 28.17881732$

Step 5.3.5

Multiply

- 1

$- 1$ by

- 28.17881732

$- 28.17881732$ .

t = 28.17881732

$t = 28.17881732$

t = 28.17881732

$t = 28.17881732$

t = 28.17881732

$t = 28.17881732$