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$\log^2n$ is what I need assistance with. How is this read in word form?

What exactly does this mean? No matter how much I read about logarithms, they still seem new to me.

Jeffrey Young
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  • One typically has that $\log^2 x$ means $(\log x)^2$. In general, $f^n(x)$ means $(f(x))^n$. – Gahawar Oct 04 '14 at 04:41
  • There is a conjecture that there is always a prime between $n$ and $n+C\log^2n$. So my question is what does $log^2n$ mean and how is it said in word form? – Jeffrey Young Oct 04 '14 at 04:53
  • Personally, to avoid ambiguity, I would say, "The square of the logarithm of $n$". – Gahawar Oct 04 '14 at 04:56
  • $\log^2 n$ just means $(\log n)^2$, or "The square of log n" (that's how I belive you should say it). – Anonymous Computer Oct 04 '14 at 05:15
  • @JeffreyYoung: In that conjecture it definitely means $(\log(n))^2$. – Marc van Leeuwen Oct 04 '14 at 05:42
  • @Gahawar: You may be right about $\log$, but in general $f^n(x)$ means $f$ applied $n$ times to $x$: $f(f(...f(x)...))$ with $n$ occurrences of $f$. The point is that writing $f(x)^n$ explicitly is easy, but for $f(f(...f(x)...))$ it is quite cumbersome. – Marc van Leeuwen Oct 04 '14 at 05:49
  • @MarcvanLeeuwen: Is the conjecture that there is ways a prime between $n$ and $n+C\log^2n$ the same as Cramér's conjecture, that there is always a prime between $n$ and $n+O((\log n)^2)$?

    Is this conjecture somehow derived from the prime counting function? I have asked about this in a new question: http://math.stackexchange.com/questions/958053/the-conjecture-that-there-is-always-a-prime-between-n-and-nc-log2n

     – Jeffrey Young Oct 04 '14 at 17:21

3 Answers3

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$\log(x)$ is a function.

Generally, the default is to use the natural logarithm or $\ln(x) = \log_e(x)$, but it is not unnatural in some settings to think of $\log(x)$ as base $10$.

The function generates a number that represents what power the base would have to be raised to to equal x. In other terms, $ log_a(x) = b $ where $a^b = x$.

For example, $2^3 = 8$, so $log_2(8) = 3$.

Finally, $\log^2(x) = (\log(x))^2$, which is just the square of the function.

  • Normally if $f$ is a function $f^2(x)$ does not mean $f(x)^2$ but $f(f(x))$ which is something quite different. But for some special function like $\sin$ and $\cos$ a different convention is used, which may be related to the fact that they are often written without parentheses as $\sin x$. – Marc van Leeuwen Oct 04 '14 at 05:35
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It is essentially the square of $\log n$. Just like $\sin^2\theta$ is $(\sin\theta)^2$, $\log^2n:=(\log n)^2$. Simply read it as "log squared of n".

Ryan Unger
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You could probably read it just about anyway you want. But, I am assuming this is the square of the Log of n so I would read it as "Log n squared" noting of course that this is slightly ambiguous. Another way would be to say "log squared n" which is OK as it sounds similar to something like $\sin^2x$.

K7PEH
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