Why is it that I never see equations/laws in STEM always using ln(x) and never laws using other log bases
In: Mathematics
I’d gonna pretend your 15, because 5 year-olds don’t know what e is.
It’s due to the definition of ln(x) and e^x, and in particular the derivative of e^x. e is defined (at least it’s one way to define it) by being the number e such that the derivative of e^x is equal to itself, e^x. The derivative of a^x for every other number a is a multiple of itself, c*a^x.
Extend this, and you just use base e for your logs and exponents, because otherwise you’d have to have an extra multiplier in every formula where you cared about rates of change, etc.
You can think of it like why we use radians rather than degrees – in general it doesn’t matter, the principles remain the same, but it makes some of the maths easier as you don’t need to remember annoying differentiation contestants like pi/180 or log_10(e) or whatever. Because e is a special number, d/dx (ln(x)) = 1/x, whereas if you used log_10(x) then you’d have a bonus constant factor.
When you first meet logs you’re pretty much just doing calculations with them on a calculator, so you might as well use whatever base is convenient.
When you get a bit deeper into the topic you find that *ln(x)* is generally easier to work with (as with *e^?*). Sometimes if you are working in a different base you’ll just have to convert it to *e* at some point to make things work.
It is a big enough deal that once you get far enough into maths you stop seeing even *ln(x)* and just see *log(x)*.
It is kind of like how the further you get into STEM subjects the more you default to radians over degrees (or gradians). The easy maths is the same, the harder maths is easier.
The big thing with *e* and *ln* is differentiation/integration, where if you’re working in some other base you have to throw in some extra constants.
The function f(x) = g^x can be written as f(x) = g^x = (e^ln(g) )^x = e^ln(g)x . In other words, any exponential function can be written with the Euler’s number e. It is a useful standardisation to make, since exponential functions can be compared more easily that way and the derivative can be computed more easily.
Of course, that means you should also use ln(x), since that’s its inverse function.
dude ln is just super chill you know. it makes math easier in calc and stuff. other logs are cool too but ln is like the go-to party trick for math nerds.
You can show that
log_a(x) = ln(x) / ln(a)
So a logarithmic expression in any base can be rewritten using the natural log. The natural of has some advantages (it’s very easy to do calculus with), so it’s the most commonly used.
But it’s not true that other bases are never used. The base-2 log is used for half-lives in physics and doubling times in biology, the base-10 log comes up in audio and electronics (decibels), geophysics (Richter scale), and chemistry (pH), and the base 100 logarithm is used in astrophysics (stellar magnitudes).
Latest Answers