Problem 13 - Precalculus: A Concise Course

Question

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$

Step-by-Step Solution

Answer

The solution to this exercise are the equations: (a) \(P = A / e^{rt}\), and (b) \(t = ln(A/P) / r\).

Step 1: Solve for P

To solve for \(P\), the task is to isolate \(P\) on one side of the equation. To do this, both sides of the equation are divided by \(e^{rt}\), giving: \(P = A / e^{rt}\).

Step 2: Solve for t

To solve for \(t\), a logarithm is necessary to remove the exponent on the variable that the task is to solve for. First, isolate the exponent, then use the property that the natural log of \(e^{x}\) is \(x\) to get rid of the exponent, and finally, apply the natural logarithm to both sides of the equation to obtain: \(ln(A/P) = rt\). After that, t can be isolated by dividing both sides of the equation by \(r\), leading to: \(t = ln(A/P) / r\).