Calculus Examples

$6000 = 3000 e^{(0.04) (t)}$

Step 1

Rewrite the equation as $3000 e^{(0.04) (t)} = 6000$ .

$3000 e^{(0.04) (t)} = 6000$

Step 2

Simplify.

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Step 2.1

Multiply $0.04$ by $t$ .

$3000 e^{0.04 (t)} = 6000$

Step 2.2

Multiply $0.04$ by $t$ .

$3000 e^{0.04 t} = 6000$

Step 3

Divide each term in $3000 e^{0.04 t} = 6000$ by $3000$ and simplify.

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Step 3.1

Divide each term in $3000 e^{0.04 t} = 6000$ by $3000$ .

$\frac{3000 e^{0.04 t}}{3000} = \frac{6000}{3000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3000 e^{0.04 t}}{3000} = \frac{6000}{3000}$

Step 3.2.1.2

Divide $e^{0.04 t}$ by $1$ .

$e^{0.04 t} = \frac{6000}{3000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $6000$ by $3000$ .

$e^{0.04 t} = 2$

Step 4

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

$\ln (e^{0.04 t}) = \ln (2)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{0.04 t})$ by moving $0.04 t$ outside the logarithm.

$0.04 t \ln (e) = \ln (2)$

Step 5.2

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

$0.04 t \cdot 1 = \ln (2)$

Step 5.3

Multiply $0.04$ by $1$ .

$0.04 t = \ln (2)$

Step 6

Divide each term in $0.04 t = \ln (2)$ by $0.04$ and simplify.

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Step 6.1

Divide each term in $0.04 t = \ln (2)$ by $0.04$ .

$\frac{0.04 t}{0.04} = \frac{\ln (2)}{0.04}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $0.04$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{0.04 t}{0.04} = \frac{\ln (2)}{0.04}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (2)}{0.04}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (2)}{0.04}$

Step 6.3.2

Log base $2.71828182$ of $2$ is approximately $0.69314718$ .

$t = \frac{0.69314718}{0.04}$

Step 6.3.3

Divide $0.69314718$ by $0.04$ .

$t = 17.32867951$