relationship between doubling time and growth rate

I have been trying to plot the growth rate in new daily COVID-19 cases (not cumulative) over time for a country. I have smoothed out the daily cases using a 7-day moving average. I then take the natural log difference between to the successive days, and consider that to be the growth rate, $r$, and plotted them as seen below

My question arises when tryin to calculate the doubling time ($T_d$) from this growth rate. I am not sure which formula to use.

If $r$ is the intrinsic growth rate, then doubling time, $T_d = \dfrac{\ln(2)}{r}$. However, there is also a formula which shows that $T_d = \dfrac{\ln(2)}{\ln(1+r)}$

Which formula should I used, given the way I have calculated growth rate?

Here is the link to the excel spreadsheet

Solution 1:

The formula $T_d=\frac{\ln(2)}{\ln(1+r)}$ is the exact doubling time under a constant discrete growth rate $r$ satisfying $\frac{y_{t+1}-y_t}{y_t}=r$, which implies $y_t=y_0(1+r)^t$.

The formula $T_d=\frac{\ln(2)}{r}$ is the exact doubling time under a constant continuous growth rate $r$ satisfying $\frac{dy/dt}{y}=r$, which implies $y_t=y_0e^{rt}$.

The two will be very close for $r$ small since $\ln (1+r)\approx r.$