Finite Math Examples

$6 = 2 e^{0.03 (t)}$

Step 1

Rewrite the equation as $2 e^{0.03 (t)} = 6$ .

$2 e^{0.03 (t)} = 6$

Step 2

Multiply $0.03$ by $t$ .

$2 e^{0.03 t} = 6$

Step 3

Divide each term in $2 e^{0.03 t} = 6$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 e^{0.03 t} = 6$ by $2$ .

$\frac{2 e^{0.03 t}}{2} = \frac{6}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 e^{0.03 t}}{2} = \frac{6}{2}$

Step 3.2.1.2

Divide $e^{0.03 t}$ by $1$ .

$e^{0.03 t} = \frac{6}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $6$ by $2$ .

$e^{0.03 t} = 3$

Step 4

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

$\ln (e^{0.03 t}) = \ln (3)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{0.03 t})$ by moving $0.03 t$ outside the logarithm.

$0.03 t \ln (e) = \ln (3)$

Step 5.2

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

$0.03 t \cdot 1 = \ln (3)$

Step 5.3

Multiply $0.03$ by $1$ .

$0.03 t = \ln (3)$

Step 6

Divide each term in $0.03 t = \ln (3)$ by $0.03$ and simplify.

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Step 6.1

Divide each term in $0.03 t = \ln (3)$ by $0.03$ .

$\frac{0.03 t}{0.03} = \frac{\ln (3)}{0.03}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $0.03$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{0.03 t}{0.03} = \frac{\ln (3)}{0.03}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (3)}{0.03}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (3)}{0.03}$

Step 6.3.2

Log base $2.71828182$ of $3$ is approximately $1.09861228$ .

$t = \frac{1.09861228}{0.03}$

Step 6.3.3

Divide $1.09861228$ by $0.03$ .

$t = 36.62040962$