Precalculus Examples

$42000 = 21000 e^{0.045 t}$

Step 1

Rewrite the equation as $21000 e^{0.045 t} = 42000$ .

$21000 e^{0.045 t} = 42000$

Step 2

Divide each term in $21000 e^{0.045 t} = 42000$ by $21000$ and simplify.

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Step 2.1

Divide each term in $21000 e^{0.045 t} = 42000$ by $21000$ .

$\frac{21000 e^{0.045 t}}{21000} = \frac{42000}{21000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $21000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{21000 e^{0.045 t}}{21000} = \frac{42000}{21000}$

Step 2.2.1.2

Divide $e^{0.045 t}$ by $1$ .

$e^{0.045 t} = \frac{42000}{21000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $42000$ by $21000$ .

$e^{0.045 t} = 2$

Step 3

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

$\ln (e^{0.045 t}) = \ln (2)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{0.045 t})$ by moving $0.045 t$ outside the logarithm.

$0.045 t \ln (e) = \ln (2)$

Step 4.2

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

$0.045 t \cdot 1 = \ln (2)$

Step 4.3

Multiply $0.045$ by $1$ .

$0.045 t = \ln (2)$

Step 5

Divide each term in $0.045 t = \ln (2)$ by $0.045$ and simplify.

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Step 5.1

Divide each term in $0.045 t = \ln (2)$ by $0.045$ .

$\frac{0.045 t}{0.045} = \frac{\ln (2)}{0.045}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.045$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.045 t}{0.045} = \frac{\ln (2)}{0.045}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (2)}{0.045}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (2)}{0.045}$

Step 5.3.2

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

$t = \frac{0.69314718}{0.045}$

Step 5.3.3

Divide $0.69314718$ by $0.045$ .

$t = 15.40327067$