Finite Math Examples

$\frac{33}{200} = e^{0.3 t}$

Step 1

Rewrite the equation as $e^{0.3 t} = \frac{33}{200}$ .

$e^{0.3 t} = \frac{33}{200}$

Step 2

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

$\ln (e^{0.3 t}) = \ln (\frac{33}{200})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{0.3 t})$ by moving $0.3 t$ outside the logarithm.

$0.3 t \ln (e) = \ln (\frac{33}{200})$

Step 3.2

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

$0.3 t \cdot 1 = \ln (\frac{33}{200})$

Step 3.3

Multiply $0.3$ by $1$ .

$0.3 t = \ln (\frac{33}{200})$

Step 4

Divide each term in $0.3 t = \ln (\frac{33}{200})$ by $0.3$ and simplify.

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Step 4.1

Divide each term in $0.3 t = \ln (\frac{33}{200})$ by $0.3$ .

$\frac{0.3 t}{0.3} = \frac{\ln (\frac{33}{200})}{0.3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.3 t}{0.3} = \frac{\ln (\frac{33}{200})}{0.3}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{33}{200})}{0.3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{33}{200})}{0.3}$

Step 4.3.2

Divide $33$ by $200$ .

$t = \frac{\log_{2.71828182} (0.165)}{0.3}$

Step 4.3.3

Log base $2.71828182$ of $0.165$ is approximately $- 1.8018098$ .

$t = \frac{- 1.8018098}{0.3}$

Step 4.3.4

Divide $- 1.8018098$ by $0.3$ .

$t = - 6.00603268$