I use both of these formulas in my financial math class, so I'll try to explain what's going on to the best of my ability.
Your first formula describes the rate of change in the account over a set period of time, expressed as a percentage of the beginning-of-period value. If you think of it in terms of how a bank works, while the bank is technically depositing small amounts of money into your account due to interest, the compounding interest just tells you what the account will be at by the end of the period. What it doesn't tell you is the exact rate of growth at any point of time
The second formula is for things that actually compound continuously, meaning they are continuously growing. Your r in that formula is something called a constant force of interest and it actually measures the instantaneous change in the account value, but expressed as an annualized percent of the current value. Suppose instead of money deposits, we think about those bacteria you mentioned. Now, the bacteria may double every month, but that doesn't mean that they grow at some set rate. They may growth extremely fast during the first three weeks and then very slowly during the last week, or vice versa, but they'll still be doubled by the end. It stands that this r is not the same as the r from the first formula, so let's call it i and it's formula is
i = ln(1+r)
So, if for example, let r be an annual effective interest rate equal to 5%. Then i = 4.879%, meaning while the account will grow by 5% at the end of the year, at any giving time, it's growing by 4.879% of what it was just a moment before.
So, if you just let i = ln(1+r) or r = ei -1, and choose the correct time, either formula should be fine.