# 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?

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 $$\ln (1+r)\approx r.$$