Exponential Growth by Doubling Time

This is exponential growth written in the parameter people actually measure. Instead of a percentage per period, it uses $T$, the time to double, and the exponent $t/T$ then simply counts how many doublings have gone by: $N = N_0 \cdot 2^{t/T}$. Three doublings is eight times, ten doublings is a bit over a thousand times, twenty is a million. Reading the exponent as a count of doublings makes the arithmetic something you can do in your head, which the percentage form never quite allows.

A worked instance. E. coli in favourable conditions divides about every 20 minutes. Left alone for eight hours that is 24 doublings, so a single cell becomes $2^{24} \approx 16.8$ million — which is the reason a water sample must be refrigerated and processed within hours, and why a warm sample line invalidates a count. Moore's law is the same equation at the other end of the time scale, with $T$ around two years.

The bridge to a percentage rate is the Rule of 70. Since $T = \ln 2 / \ln(1+r) \approx 0.693/r$ for small $r$, dividing 70 by the percentage growth per period gives the doubling time closely enough for mental arithmetic: 3.5% a year doubles in about twenty years, 7% in about ten. Bankers use 72 instead of 70 because it divides more conveniently and is slightly more accurate over the range of interest rates they care about. Note also that only the ratio $t/T$ enters, so the answer is unitless in time — any clock works, provided both use the same one.

That proviso is the first mistake: $t$ in hours with $T$ in minutes gives an exponent sixty times too large, and the result will be absurd rather than merely wrong, which at least makes it catchable. The second is a failure of intuition rather than arithmetic. Exponential growth does not feel fast until suddenly it is overwhelming, because each doubling adds more than everything that came before it combined. The pond that is fully covered by lilies on day thirty was half bare on day twenty-nine. The third is the honest limit: nothing doubles forever. Bacteria exhaust their substrate, markets saturate, transistors hit atomic scales. Every exponential in the real world is the early portion of an S-curve, and this formula describes only the part before the ceiling shows up. Using it to project far ahead is not a calculation so much as an assumption that nothing will run out.