The discussion clarifies the notation surrounding logarithmic functions, specifically the distinction between ##\log^3 n## and ##\log^{(3)} n##. The former represents the cube of the logarithm, or ##(\log n)^3##, indicating repeated multiplication, while the latter denotes the composition of the logarithm applied three times, or ##\log(\log(\log n))##. Participants emphasized the importance of clear notation to avoid confusion with derivatives, as similar symbols can represent different mathematical concepts.
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That's the one you want.22990atinesh said:or ##\log^3 n = (\log n)^3 ##
HallsofIvy said:It is not uncommon to put a an exponent in parentheses to indicate a repeated composition.
That is, (log(x))^{(3)} or log^{(3)}(x) is "log(log(log(x)))".
Unfortunately, that is also often used to indicate the third derivative so you must be careful to state which!
Mark44 said:In exactly the same way, cos2(x) does not mean cos(cos(x)), but rather it means cos(x) * cos(x) = (cos(x))2. The exponent indicates repeated multiplication, not repeated function composition.
The above is what HallsOfIvy said. I haven't seen it, myself, but I have seen f(3), with parentheses around the exponent, to indicate the third derivative.22990atinesh said:You mean ##\log^{(3)} n = (\log n)^{(3)} = \log (\log(\log n))##
This is what I said.22990atinesh said:##\log^3 n = (\log n)^3##
I know, I was just rechecking from you. :)Mark44 said:The above is what HallsOfIvy said. I haven't seen it, myself, but I have seen f(3), with parentheses around the exponent, to indicate the third derivative.
This is what I said.