dieter_hermann

What is so special about the natural logarithm? What is it good for?

https://feddit.de/post/1055636

What is so special about the natural logarithm? What is it good for? - Feddit

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?

Jun 24, 2023 at 9:42amWeb
Show threadkiwifoxtrotJun 24, 2023
ln(x) and e^x are to simulate growth rates, which is why I assume it's called the natural log. ln(x) will give you the time it will take to grow your population to size x. e^x will give you the size of your population after x time assuming 100% growth rate. This works really well for bacteria as they divide into two to grow, 1 -> 2 -> 4 -> 8 etc.
Show threadrbhfdJun 24, 2023
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!
Show threadgreen_pyroxeneJun 24, 2023
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.
Show threadperson594Jun 24, 2023

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).

Show threaddieter_hermannJun 24, 2023

Your explanation makes a lot of sense to me! I didn't know that exp'(x) = exp(x), but can see how this could be an interesting property and in turn makes ln interesting.

You saying, that it often doesn't matter which logarithm is used, made me check and realize that log_a(x)/log_a(y) is the same as log_b(x)/log_b(y). Thus I understand that it really doesn't make a difference which logarithm is used when "comparing the magnitude" (not sure if this is the right term) of numbers.

I feel like I have a much better understanding of ln now. I'll assume that the base of an algorithm is often basically a random choice and base e is often used because of its "interesting trivia".

Thanks a lot!

Show threadsomeacnt_Jun 24, 2023
One word: differentiation.
Show threadx7123m3_256Jun 24, 2023

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.

Show threaddieter_hermannJun 24, 2023
The simple derivative really is nice, thanks for pointing this out!
Show threadAwkward_PenguinJun 24, 2023
I highly recommend checking out these articles from Better Explained. It's what made it finally click for me.
An Intuitive Guide To Exponential Functions & e – BetterExplained

Show threadleptiricaJun 24, 2023
Didn't think I'd stay up late reading about math, but here we are.