Why Does the Rule of 72 Work? The Real-World Use of Logarithms
Many people have come across a quick rule of thumb in personal finance books or talks: divide 72 by the annual interest rate, and you get an approximate number of years needed for your money to double. For example, at an annual interest rate of 6%, 72÷6=12, so the principal will roughly double in 12 years; at 8%, it takes about 9 years.
The “Rule of 72” is easy to remember and useful in practice. Yet few students stop to ask: why 72? And when does the rule become inaccurate?
These two questions provide a very practical application of exponents and logarithms in senior secondary mathematics. Students often ask, “What is the use of logarithms?” This is one very real answer.
Start with the exact equation
Suppose an initial sum grows annually at a compound interest rate of r. If it doubles after n years, then:
(1 + r)ⁿ=2
The unknown appears as an exponent, which is exactly where logarithms become useful. Taking natural logarithms on both sides gives:
n=ln 2÷ln(1+r)
For an annual interest rate of 6%,
n＝ln 2÷ln 1.06≈11.9
So the money doubles in about 11.9 years—very close to the 12 years predicted by the Rule of 72.
Of course, nobody wants to calculate natural logarithms while standing at a bank counter. The value of the Rule of 72 is that it turns this formula into something that can be estimated mentally.
Where does 72 come from?
ln 2 is approximately 0.693. When r is small, ln(1 + r) is approximately equal to r itself. This is the linear approximation of the curve y = ln(1 + x) near x = 0, which students encounter in the calculus component of IB Mathematics. Therefore, n ≈ 0.693 ÷ r. If the interest rate is expressed as a percentage R—for example, 6% means R = 6—then n ≈ 69.3 ÷ R.
So why use 72 rather than 69.3? There are two reasons. First, ln(1 + r) is in fact slightly smaller than r, so the true doubling time is slightly longer than 69.3 ÷ R. Increasing the numerator slightly makes the estimate more accurate within the common interest-rate range of around 5% to 10%. At an annual interest rate of about 8%, the Rule of 72 is almost exact.
Second, 72 is easy to calculate mentally because it is divisible by 2, 3, 4, 6, 8, 9, and 12. In other words, 72 is not a mathematical theorem, but a practical compromise between accuracy and convenience.
Interestingly, if interest is compounded continuously rather than annually, the exact doubling-time formula is n = ln 2 ÷ r, which is approximately 69.3 ÷ R when the rate is expressed as a percentage. For the same question, changing the assumption changes the most suitable number to use.
When does it become inaccurate?
The table below compares the Rule of 72 with the exact doubling time.
At very low interest rates, the Rule of 72 slightly overestimates the doubling time. As the interest rate rises, it begins to underestimate it; at 36%, the error exceeds 10%.
The graph below shows that this is not a random pattern. The error changes systematically with the interest rate, because the approximation ln(1+r)≈r gradually becomes less accurate.
The error in the Rule of 72 changes with the annual interest rate. It is most accurate at around 8%; as the interest rate rises, the estimated doubling time becomes increasingly too short.
More than just personal finance
The same equation applies to any quantity that grows or declines by a fixed percentage.
In economics, if prices rise by 3% each year, the purchasing power of money halves in around 23 to 24 years. If an economy grows by 7% annually, its output roughly doubles in 10 years. Bacterial reproduction in biology and population growth in geography are also described using doubling time.
The “72” that students encounter across different subjects is, in fact, based on the same logarithmic question.
Knowing how to use a tool also means knowing when not to use it
Applying the Rule of 72 is not difficult. The harder—and more important—task is being able to explain its assumptions and identify when it starts to fail. This judgement of knowing when a tool can be trusted is more valuable than simply memorising a formula. It is also one of the abilities that senior secondary mathematics aims to develop.
Parents do not need to understand logarithms to ask their children three questions:
How different is the answer from the Rule of 72 compared with the exact answer? How do you know?
If the annual interest rate were 30%, would you still use this rule? Why or why not?
Apart from saving money, where have you seen the same type of growth in other subjects?
If a child can answer these questions clearly, it shows that they have not merely memorised a formula—they truly understand it.
Written by: Ken Sir
Original Article: HKEJ Education
English translation provided by AI.
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