The equation for compound interest is usually given in the form P=A(1+r)^n, where A is the initial amount, P the amount after n time periods and r the compound interest rate for the time period. n=12T if the interest is compounded monthly and T is in years, with r as the monthly rate of interest. Example: if r=4.8% per annum (0.048) then r=0.4% per month (0.004).
This equation can be rewritten: P/A (growth)=(1+r)^(12T), so ln(growth)=12Tln(1+r) and T=ln(growth)/(12ln(1+r)), which resembles the given equation, in which growth is 3.17 and r is the rate per annum, making 0.08333r (=r/12) the rate per month. T(r)=ln(3.17)/(12ln(1+0.08333r)).
Clearly, if r=0 there would be no growth at all and T would be infinite, hence r>0. This equation relates T and r and would be the basis of a table where the growth was fixed at 3.17 or 317%. Such a table appears below:
| r (rate % per annum) |
T years |
| 1 |
115.42 |
| 2 |
57.73 |
| 3 |
38.51 |
| 4 |
28.89 |
| 5 |
22.12 |
| 6 |
19.28 |
| 7 |
16.53 |
| 8 |
14.47 |
| 9 |
12.87 |
| 10 |
11.59 |
A graph based on the table is also useful. The equation can't be "solved" for either T or r, since we would need either T or r to find r or T.