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.
One word: differentiation.
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!
So to be honest, 90% of the time the base of the logarithm doesn’t really matter as long as we are consistent. The main property we use logarithms for is that log_b(xy) = log_b(x) + log_b(y), and this holds for any base b. In fact, the change-of-base formula tells us that we can get from one base to another just by multiplying by a constant (log_a(x) = log_b(x) * 1/log_b(a)), and so there is a strong desire to pick one canonical “logarithm” function, and just take care of any base silliness by multiplying your final result by a scaling factor if needed.
Given that, the natural logarithm is quite “natural” because it is the inverse of the exponential function, exp(x) = e^x. The exponential function itself is quite natural as it is the unique function f such that f(0) = 1 and f’(x) = f(x). Really, I would argue that the function exp(x) is the fundamentally important mathematical object – the natural logarithm is important because it is that function’s inverse, and the number e just happens to be the value of exp(1).