I understand that log2 is useful for informatics, e.g. to determine how many bits I need to represent a given number. I understand that log10 is useful to determine the order of magnitude of numbers.
However, I’m having trouble understanding what makes ln interesting. It seems like it’s used a lot, but to me it just looks like a logarithm with a very weird base. What are the uses for this logarithm?
It’s the inverse of the natural exponential function. So if you want to solve something like e^x=49 you can apply ln to both sides and get X=ln(49). e is a very natural constant for various reasons. For example suppose that something doubles each year, which you could express as something like y = x*2. Now suppose that you want do distribute that growth over multiple steps each year. So it would for example increase by 50% each half year. So for a year you’d get two steps: y= x*1.5*1.5 = x*2.25. Which is more because of compound interest.
If you increase these intermediate steps towards infinity, that factor becomes e.