The rule of 72 is a quick way to evaluate approximately how long it will take for your money to double given a specific interest rate R.
Say for example, if a bank offers you an annual interest rate of 5%, then it will take approximately 72/5 =14.4 years for your initial deposit to double in value, assuming the 5% holds throughout the whole period and no withdrawals are made by you.
It's an simple rule crafted to aid in mental math:
Suppose we want to solve how many compounding periods (usually annual) it takes to double our money, it becomes solving for T (time) given initial money A and interest rate X, we thus have:
2A = A(1 + X/100)^T
2 = (1 + X/100)^T
Take natural log (In) on both sides, we have
T = In(2)/ In(1 + X/100),
which is an expression close to 72/X. To illustrate how close they are, I have plotted both functions on Desmos:
Notice how close both curves are and only start to diverge when X (interest rate) hits 15.
As mentioned above, the rule of 72 is a quick mental aid. People have argued that using 69 or 70 is even more accurate, but 72 has taken off primarily because it's easier to do the division I guess. You are free to use 69, 70 or 72 this is merely serving as a quick mental guide, it's not meant to help you reach an accuracy level up to the 17th decimal place.
P.S: Would appreciate it if any reader can introduce me a blogsite where I can type Mathematics the way Mathtype, MS equations or Latex allows.