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What is it used for and why doesn't it equal $\large\log{x^2}$?

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It's shorthand for $(\log x)^2$. The problem is that $\log (x^2)=2\log x$ by the power rule for logarithms, and in general $y^2\ne 2y$ (here with $y=\log x$). – anon May 27 '12 at 23:25
up vote 12 down vote accepted

It is convention to write

$$\log ^2 x$$

in place of

$$(\log x)^2=\log x \times \log x$$

just in the same manner as we write $\sin^2 x = (\sin x)^2$ or any other "named" function's square. In general, you will see $(f(x))^n$ for the $n$th power of a function, and $f^n (x)$ in place of the $n$ times composition of $f$, i.e. $f^3(x)=f \circ f \circ f(x)$. However, for the $\log$ and trigonometric functions, we break this convention.

Note that $\log(x^2)=2 \log x \neq \log x \cdot \log x$

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Great answer. This cleared up everything and taught me a bit too! :) – David May 27 '12 at 23:27
Barring special cases like $\sin$, $f^n(x)$ is rather rare notation for this, presumably because the more precise $f(x)^n$ is convenient with parenthesized evaluation. – Hurkyl May 28 '12 at 1:15
Similar to @Hurkyl, in my experience it is most common for $f^n(x)$ to mean the $n$-times application of $f$ , and the interpretation of raising $f(x)$ to the $n$th power is only used for trigonometric functions and $\log$. – Chris Taylor May 28 '12 at 9:50

Already "log" is ambiguous, implying respectively base 10, e, or 2 in elementary applied mathematics, general mathematics, and information theory or theoretical computer science. In the second case, particularly in number theory and theoretical statistics, the iterated logarithm arises naturally, and some authors mean $\log \log x$ by $\log^2 x$. The squared logarithm isn't seen much outside school calculus textbooks. The reverse is true for trigonometric functions: the squared functions are ubiquitous, while the iterated functions are mostly confined to examples and exercises for students. The notation $\sin^2 x$ is illogical (Gauss, in particular, complained about it), but so convenient and established by tradition that we are probably stuck with it.

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Both squared and iterated logarithms arise naturally in algorithmic complexity, and there the notation $\log^2 n$ for $(\log n)^2$ is ubiquitous. Iterated logarithms are notated $\log \log n$. – Henning Makholm Apr 6 '15 at 5:22

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