Finite Math Examples

$2000 = 4800 e^{0.055 t}$

Step 1

Rewrite the equation as $4800 e^{0.055 t} = 2000$ .

$4800 e^{0.055 t} = 2000$

Step 2

Divide each term in $4800 e^{0.055 t} = 2000$ by $4800$ and simplify.

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Step 2.1

Divide each term in $4800 e^{0.055 t} = 2000$ by $4800$ .

$\frac{4800 e^{0.055 t}}{4800} = \frac{2000}{4800}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4800$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4800 e^{0.055 t}}{4800} = \frac{2000}{4800}$

Step 2.2.1.2

Divide $e^{0.055 t}$ by $1$ .

$e^{0.055 t} = \frac{2000}{4800}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2000$ and $4800$ .

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Step 2.3.1.1

Factor $400$ out of $2000$ .

$e^{0.055 t} = \frac{400 (5)}{4800}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $400$ out of $4800$ .

$e^{0.055 t} = \frac{400 \cdot 5}{400 \cdot 12}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{0.055 t} = \frac{400 \cdot 5}{400 \cdot 12}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{0.055 t} = \frac{5}{12}$

Step 3

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

$\ln (e^{0.055 t}) = \ln (\frac{5}{12})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{0.055 t})$ by moving $0.055 t$ outside the logarithm.

$0.055 t \ln (e) = \ln (\frac{5}{12})$

Step 4.2

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

$0.055 t \cdot 1 = \ln (\frac{5}{12})$

Step 4.3

Multiply $0.055$ by $1$ .

$0.055 t = \ln (\frac{5}{12})$

Step 5

Divide each term in $0.055 t = \ln (\frac{5}{12})$ by $0.055$ and simplify.

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Step 5.1

Divide each term in $0.055 t = \ln (\frac{5}{12})$ by $0.055$ .

$\frac{0.055 t}{0.055} = \frac{\ln (\frac{5}{12})}{0.055}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.055$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.055 t}{0.055} = \frac{\ln (\frac{5}{12})}{0.055}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{5}{12})}{0.055}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{5}{12})}{0.055}$

Step 5.3.2

Divide $5$ by $12$ .

$t = \frac{\log_{2.71828182} (0.41\overline{6})}{0.055}$

Step 5.3.3

Log base $2.71828182$ of $0.41\overline{6}$ is approximately $- 0.87546873$ .

$t = \frac{- 0.87546873}{0.055}$

Step 5.3.4

Divide $- 0.87546873$ by $0.055$ .

$t = - 15.9176134$