The amount A in an account after t years from an initial principle P invested at an annual rate r compounded c

Question

The amount A in an account after t years from an initial principle P invested at an annual rate r compounded continuously is given by A = Pe^rt where r is expressed as a decimal. Solve this formula for t in terms of A, P, and r.
a. t = ln(AP/r)
b. t = ln(A/rP)
c. t =r ln(A/P)
d. t=1/r ln(A/P)

Answer 1

A = Pe^rt ---------- divide both sides by P to get
A/P = e^rt --------- take the natural log of both sides to get
ln (A/P) = ln(e^rt) --------- simplify right side to get
ln (A/P) = rt*ln(e) ---------simpllify right side further to get
ln (A/P) = rt ------- divide both sides by r to get
(1/r)ln (A/P) = t

The answer is D. . .

Answer 2

A = Pe^(rt)
A/P (growth as a percentage of principle) = e^(rt)
ln(A/P) = rt
[ ln(A/P)]/r = t

The "rule of 72" is relevant here. If A/P is 2, doubling your investment, 72/r (r as a percent without a decimal) gives you roughly the time required. For example, at 8% you double your investment in 72/8 = 9 years. Business types use 72 because it's close to ln(2) = 0.6931 and has lots of nice factors like 6, 4, 3, and so on.