26 March 2020

Exponential growth - Year 10 maths revision

"When will I ever need to use logarithms, or raising something to the the power of 1/n in real life". During Covid-19, it turns out. Since no one likes logarithms, I thought I'd just post a few formulas. The following should work on any system of exponential growth.

(Naturally, for Excel formulas, substitute in the value or cell reference for X, Y, T, etc)

FormulaExcel FormulaExample
Initial CasesX112  Aus # Mar10
Final CasesY2,431  Aus # Mar25
TimeT15  days
Multiplier over durationM = Y/X=Y/X21.71
Daily MultiplierMD = M1/T=(Y/X)^(1/T)1.23
Daily % Increase    =(MD -1) * 100=((Y/X)^(1/T)-1)22.8%  Format as %
Time to doubleT2X = loge2 / logeMD=LN(2)/LN((Y/X)^(1/T))3.38  days
Time to 10xT10X = loge10 / logeMD=(Y/X)^(1/T)/LN(10)11.22  days
Convert from 'time to double' to 'time to 10x'
T10X = T2X * loge10 / loge2=TDOUBLE * LN(10)/LN(2)
Convert from 'time to 10x' to 'time to double'
T2X =T10X * loge2 / loge10=TTEN * LN(2)/LN(10)
Convert from 'time to double' to 'daily multiplier'
MD = elog2 / T2X=EXP( LN(2) / TDOUBLE)
Apply a daily multiplier for N days

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