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 not a weird base, it’s really the most natural base to choose, which is why it’s called the natural logarithm. It doesn’t particularly matter what base you choose, because you can always convert from one base to another, but often the natural logarithm is simpler to work with. For example, the derivative of ln(x) is just 1/x. The derivative of log10(x) is 1/(x*ln(10)).
This is because ln(x) is the inverse of e^x, which has the unique property that it is its own derivative.
The simple derivative really is nice, thanks for pointing this out!
The ‘e’ base is an interesting choice in calculus. For example, the derivative of ln(x) wrt x is 1/x. The derivative of log10(x) wrt x is 1/(ln(10)x). The natural logarithm automatically pops up!
on top of that, ‘e’ is the only base b such that d/dx b^x = b^x. every other base induces some multiplicative factor in the derivative, hence why it’s considered the ‘natural’ base for exponentials and logarithms.
One word: differentiation.
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.