Trigonometry Examples

$500 = 100 e^{(0.04 t)}$

Step 1

Rewrite the equation as $100 e^{0.04 t} = 500$ .

$100 e^{0.04 t} = 500$

Step 2

Divide each term in $100 e^{0.04 t} = 500$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 e^{0.04 t} = 500$ by $100$ .

$\frac{100 e^{0.04 t}}{100} = \frac{500}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 e^{0.04 t}}{100} = \frac{500}{100}$

Step 2.2.1.2

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

$e^{0.04 t} = \frac{500}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $500$ by $100$ .

$e^{0.04 t} = 5$

Step 3

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

$\ln (e^{0.04 t}) = \ln (5)$

Step 4

Expand the left side.

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Step 4.1

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

$0.04 t \ln (e) = \ln (5)$

Step 4.2

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

$0.04 t \cdot 1 = \ln (5)$

Step 4.3

Multiply $0.04$ by $1$ .

$0.04 t = \ln (5)$

Step 5

Divide each term in $0.04 t = \ln (5)$ by $0.04$ and simplify.

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Step 5.1

Divide each term in $0.04 t = \ln (5)$ by $0.04$ .

$\frac{0.04 t}{0.04} = \frac{\ln (5)}{0.04}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.04$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.04 t}{0.04} = \frac{\ln (5)}{0.04}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (5)}{0.04}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (5)}{0.04}$

Step 5.3.2

Log base $2.71828182$ of $5$ is approximately $1.60943791$ .

$t = \frac{1.60943791}{0.04}$

Step 5.3.3

Divide $1.60943791$ by $0.04$ .

$t = 40.23594781$