Hello,
I am running panel data fixed effect regressions where health outcome counts (mortality & hospital visits) per month are explained by the number of days each month in different temperature categories. The unit of observation is municipality-month. My outcome variable is in the following form, similarly to previous literature: (outcome count in a certain group / number of people in the group * 100,000)

Until now, I have divided the coefficients of above regressions with the mean of the dependent variable in order to discuss percentage results, in the following fashion:
increase in hospital visits per 100,000 / mean of visits per 100,000
Now, I need to repeat these regressions to obtain percentage effects in order to make useful coefficient plots. I'm wondering if the correct way to do this is to take a natural log of the above expression of the outcome variable and run the regressions. The issue is that this results in largely different percentage effects to the ones obtained by dividing the coefficients by the mean of dep. var., and I wonder how this difference between the two methods might arise, and what the correct way is to obtain percentage effects. The regressions are weighted by population, could this be the issue?

Grateful for any help,
Roberto