How many doublings take a quantity up by a factor of 100, and how many years is that at 12% a year? Compare the rule of 72 answer against the exact one.

How many doublings take a quantity up by a factor of 100, and how many years is that at 12% a year? Compare the rule of 72 answer against the exact one.

Approach: Express the growth factor as a power of two using the log base 2, then convert doublings into years with a doubling time and check the shortcut against the exact logarithmic result.

6.64. A factor of 100 needs log base 2 of 100, which is 2 * log2(10) = 2 * 3.3219 = 6.6439 doublings. The check is that 2^6 = 64 and 2^7 = 128, so the answer sits about two thirds of the way between 6 and 7. At 12% a year the rule of 72 gives a doubling every 72/12 = 6 years, so a hundredfold takes about 6.6439 * 6 = 39.9 years. The exact figure is 40.6 years, from ln(100)/ln(1.12) = 4.6052/0.11333. The rule of 72 runs 1.9% fast at 12%, since the true doubling time there is 6.116 years, and that error compounds across six and a half doublings, which is why the shortcut is fine for a quote and the logarithm carries a decision.

Follow-up: At what annual rate does a hundredfold take exactly 25 years, and what does the rule of 72 predict for that rate?

Key concepts: log base 2, doubling, rule of 72.