Notation for writing log has always bugged me. Like I feel like it should be more like <10>^<527> which would be the log base 10 of 527. That's not it, but something. The current notation just doesn't feel quite right.
I never understand why anyone thinks its good notation. The layout means nothing and lets you infer nothing because exponentiation isn't a 2d planar geometric operation. All the information and rules are contained in the idea "exponentials map the additive reals to the multiplicative reals".
The notation conveys no information at all, and provides no means of proving anything, and even the notation for function composition is worse.
Given the operator pow : R^2->R there are 2 possible inverses. root and log
root isn't even interesting. its just pow with the 2nd argument precomposed with reciprocal. root x b = pow x 1/b
The Triangle of power explanation of logarithms is what really got me across logs.
It wasn't until seeing the triangle and having the relationships explained that I had any clue about logarithms, up until then logs had been some archaic number that meant nothing to me.
Because of the triangle of power, I now rock up to B and B+ Trees and calculate the number of disc accesses each will require in the worst case, depending on the number of values in each block (eg, log2(n), log50(n) and log100(n))
Ironically, that notation, which I just discovered, confuses me more than anything else.
Logs clicked for me when someone online said "amongst all the definitions we have for logs, the most useful and less taught is that log() is just a power". At that exact instant, it's like if years of arcane and foreign language just disappeared in front of my eyes to leave only obviousness and poetry.
I don't understand the comment about it just being a power, but, for me, knowing that it's filling in the third vertice on the triangle with exponents at the top, and n on the other is what makes it work for me - I now know in my head when I am looking for the log of n, I am looking for the exponent that would turn the log into n.
I don't go looking for the exact log, I only look for whole numbers when I am calculating the value in my mind.
But it makes sense when I am looking for the log2 of 8 to know that the answer is "what exponent will make 2 into 8"? and that's "3"
I think it's good for explanation purposes, but not actually great as notation, especially for operations that are so common, too much room for ambiguity, especially when writing quickly.
Well, currently exponentiation has a superscript exponent to the right of the base, and logs have a subscript base to the left of the exponent. They're already very similar to your example, but they also include the word "log".
