Notation of logarithm and its exponent I am little confused about this notation, $\log^3 n$. Does it mean $(\log n)^3$ or $\log (\log (\log n))$?
 A: As such, it means nothing, if no explicit definition has been given. It is not a standard notation, unlike notations like $\sin^k x$, which by definition means $(\sin x)^k$. It is a fair guess that $\log^3 x$ is analogously meant to denote $(\log x)^3$, but still just a guess. The superscript could alternatively be a misplaced subscript (base indicator).
Note that the identifier “log”, when used without a subscript, denotes a logarithm with an unspecified base, to be used “when the base does not need to be specified” (ref.: ISO 80000-2, clause 2-12.4), i.e. when base has been specified earlier or the text discusses properties that the logarithm functions have independently of base. It is inappropriate, and causes real risks of misunderstanding, to use “log” instead of “lg” (base 10 logarithm) or “ln” (natural logarithm) or “lb” (base 2 logarithm), expecting “log” as such to indicate a specific base.
A: $log^3n$ or $(logn)^3$ means multiplication of $log n$ three times. $log n^3$means logarithm of $n^3$. Hope this helped. 
