Now that we are getting single digit daily percent increases in our Coronavirus tracking Tables, it is time to specify even longer doubling times that they correspond to. So here are the roots of 2 up to the 15th root, and the daily percent increase, which when compounded to the power or days given, give a doubling of the initial amount.
| Days to Double. | The root of 2. | The daily percent increase |
| 1 | 2 | 100% |
| 2 | 1.414 | 41.4% |
| 3 | 1.260 | 26.0% |
| 4 | 1.189 | 18.9% |
| 5 | 1.149 | 14.9% |
| 6 | 1.126 | 12.6% |
| 7 | 1.104 | 10.4% |
| 8 | 1.0905 | 9.05% |
| 9 | 1.08o1 | 8.01% |
| 10 | 1.0718 | 7.18% |
| 11 | 1.0650 | 6.50% |
| 12 | 1.0595 | 5.95% |
| 13 | 1.0548 | 5.48% |
| 14 | 1.0508 | 5.08% |
| 15 | 1.0473 | 4.73% |
Let’s show how this is related to the slope on Logarithmic plots.
Let R(t) be a ratio of a time dependent quantity Y(t) to its initial value Y0 at time t = 0.
R(t) = Y(t)/Y0.
Let T be the doubling time in days, and t be measured in days. Then at a time t:
R(t) = 2^(t/T),
so, when t = T, R(t) = 2.
Now, using the logarithm of 2 to the base 10:
R = (10^ (log 2) )^(t/T) = 10^ ((log 2) t/T).
Taking the log of both sides,
log (R) = (log (2)/T) t. This is a line in the plot of log (R) versus t of
slope = log (2)/T = 0.30103 / T.
For T = 1 day for doubling, the slope is 0.30103
For T = 7 days for doubling, the slope is 0.30103/7 = 0.0215.
