Calculus Examples

$P = a e^{k t}$

Step 1

Rewrite the equation as $a e^{k t} = P$ .

$a e^{k t} = P$

Step 2

Divide each term in $a e^{k t} = P$ by $a$ and simplify.

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Step 2.1

Divide each term in $a e^{k t} = P$ by $a$ .

$\frac{a e^{k t}}{a} = \frac{P}{a}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $a$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a e^{k t}}{a} = \frac{P}{a}$

Step 2.2.1.2

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

$e^{k t} = \frac{P}{a}$

Step 3

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

$\ln (e^{k t}) = \ln (\frac{P}{a})$

Step 4

Expand the left side.

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Step 4.1

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

$k t \ln (e) = \ln (\frac{P}{a})$

Step 4.2

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

$k t \cdot 1 = \ln (\frac{P}{a})$

Step 4.3

Multiply $k$ by $1$ .

$k t = \ln (\frac{P}{a})$

Step 5

Divide each term in $k t = \ln (\frac{P}{a})$ by $k$ and simplify.

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Step 5.1

Divide each term in $k t = \ln (\frac{P}{a})$ by $k$ .

$\frac{k t}{k} = \frac{\ln (\frac{P}{a})}{k}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $k$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{k t}{k} = \frac{\ln (\frac{P}{a})}{k}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{P}{a})}{k}$