Approximating Exponential Growth

An investment grows 7% a year. Roughly how many years to double? You could set up $1.07^n = 2$ and solve with logs, but there's a shortcut that skips the calculator entirely and gets within a fraction of a year: the Rule of 70.

Where the rule comes from

Growth at rate $r$ (as a decimal) for $n$ years multiplies your value by $(1+r)^n$. For small $r$, $\ln(1+r) \approx r$ — a standard small-number approximation. Doubling means $(1+r)^n = 2$, i.e. $n\ln(1+r) = \ln 2 \approx 0.693$. Substituting the approximation:

$$n \approx \frac{0.693}{r} .$$

In plain English: the number of periods to double is roughly a fixed constant (about 0.693, or 69.3%) divided by the growth rate — and using 70 instead of 69.3 as the constant makes the arithmetic even easier while staying accurate for typical interview rates (roughly 1% to 20%).

Worked example 1: doubling at 7%

$$n \approx \frac{70}{7} = 10 \text{ years} .$$

Check against the exact answer: $1.07^{10} \approx 1.967$, close to 2 — the Rule of 70 slightly underestimates here but lands within a few percent, well inside what's needed for a mental estimate. The approximation gets slightly worse as $r$ grows larger, since the underlying $\ln(1+r)\approx r$ step is only exact in the limit as $r\to0$.

Worked example 2: doubling at 2% vs 20%

At 2%: $n \approx 70/2 = 35$ years. At 20%: $n \approx 70/20 = 3.5$ years. Notice the relationship is inversely proportional, not linear — going from 2% to 20% (a 10x increase in rate) cuts the doubling time by 10x too (35 down to 3.5), a direct consequence of the $n \approx 70/r$ form. This inverse relationship is worth internalizing on its own: doubling the growth rate roughly halves the time to double, and it's exactly this pattern that makes compounding at even modestly higher rates dramatically more powerful over decades.

What this means in practice

The Rule of 70 (and its cousins — Rule of 72 for rates that divide nicely by whole numbers, like 6% or 8%) is a workhorse for any quick compounding estimate: how long until a fund doubles, how fast a liability grows, or roughly how many compounding periods a strategy needs to reach a target. It's built on the same small-rate log approximation that underlies continuous compounding more generally, so understanding why it works (not just the number 70) transfers directly to other exponential-growth mental-math questions.