Calculus Examples

$b = p e^{r t}$

Step 1

Rewrite the equation as $p e^{r t} = b$ .

$p e^{r t} = b$

Step 2

Divide each term in $p e^{r t} = b$ by $p$ and simplify.

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Step 2.1

Divide each term in $p e^{r t} = b$ by $p$ .

$\frac{p e^{r t}}{p} = \frac{b}{p}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $p$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{p e^{r t}}{p} = \frac{b}{p}$

Step 2.2.1.2

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

$e^{r t} = \frac{b}{p}$

Step 3

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

$\ln (e^{r t}) = \ln (\frac{b}{p})$

Step 4

Expand the left side.

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Step 4.1

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

$r t \ln (e) = \ln (\frac{b}{p})$

Step 4.2

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

$r t \cdot 1 = \ln (\frac{b}{p})$

Step 4.3

Multiply $r$ by $1$ .

$r t = \ln (\frac{b}{p})$

Step 5

Divide each term in $r t = \ln (\frac{b}{p})$ by $r$ and simplify.

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Step 5.1

Divide each term in $r t = \ln (\frac{b}{p})$ by $r$ .

$\frac{r t}{r} = \frac{\ln (\frac{b}{p})}{r}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $r$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{r t}{r} = \frac{\ln (\frac{b}{p})}{r}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{b}{p})}{r}$