some of y’all on here: *casually doing PhD math for fun*

me: is square root a type of logarithm?

@kasdeya This is actually a really good question. We are taught that a square root is the inverse of squaring a number, and we are taught that logarithms are the inverse of exponentiation. But they are different inverses.

Technically a square root is an exponent of 1/2. So square-root(x) = x^1/2. And then it follows the same rules as exponents, and everything makes a lot more sense in terms of how they interact with other numbers. (Still have to know the rules for exponents, but at least it is less rules to remember.) So (√x)² = x^1/22 = x^2*(1/2) = x¹ = x. And the inversion is happening to the exponent, leaving you with the same base value.

A logarithm is a function that takes a base and a number and returns the exponent it would require to turn that base into that number.

So like if you would normally represent it as z = x^y, and you had z and x, then y=log_x(z).

Using 10 and 100:
Log_10(100) = 2
because 10²⁼¹⁰⁰
and Log_100(10) = 1/2
because 100^1/2 = 10

The logarithm is the inverse of applying exponentiation with a given base to a value. So it's the inverse of the function as applied to the exponent, rather than applied to the base.

Not even sure that fully clarifies the issue, it's complicated lol. Even asking the question is relatively advanced!

@kasdeya good video that highlights the symmetry https://www.3blue1brown.com/lessons/triangle-of-power

@Shivaekul thank you for the explanation! I really appreciate it. and this definitely makes sense! so it sounds like given a number x**y you can get x by doing (x**y)**1/y and you can get y by doing log_x(x**y)

also omg interesting! I didn’t even know that (x**y)**z could be simplified to x**(y*z). but I think that makes sense because

(2**2)**2 =
(2 * 2)**2 =
(2 * 2) * (2 * 2) =
2**4

and in general (x**y)**z means multiply x by itself y times, then multiply that value by itself z times. so that is just multiplying the exponents

hm I guess that means that x**y * x**z = x**(y+z). interesting that multiplication gets “downgraded” to addition and exponentiation gets “downgraded” to multiplication

@kasdeya That's exactly how it works, exponent rules too! And yeah, it is really interesting how they translate, kinda cool once you figure it out!

@spubby oohh interesting! this does look like a much clearer notation. the video went at a pace where I couldn’t follow a lot of things, but I might watch it much more slowly and pause to experiment so I can really absorb it

@kasdeya i like how 3blue1brown presents stuff, he explains things intuitively without compromising on correctness, which i find is really rare. but i do find you really have to "pause and ponder" (as he encourages) if you want to fully digest everything :)

@kasdeya rewatching the video i think he intentionally doesn't fully explain many of the relationships, since it would be more useful for you to discover that yourself