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What is the difference between $\log_2^2 x$ and $(\log_2 x)^2$? And what methods should be used for solving equations with the first case. For second I use u-substitution method in equations that have logarithm raised by different powers.

For instance,

$\log_2^2 x - \log_2 x = 2$

(We asume that $\log_2^2 x = \log_2 (log_2 x)$)

It would be nice, if you could also explain, how to solve this one.

Nyklu
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  • Please verify that my edit is correct and does not change your question. – Sarvesh Ravichandran Iyer Oct 25 '16 at 10:09
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    That depends on the used convention. $\log_2^2 x$ is frequently used to mean a) $(\log_2 x)^2$ or b) $\log_2 (\log_2 x)$. – Daniel Fischer Oct 25 '16 at 10:10
  • Can you give an example of the sort of equations you're trying to solve involving these, since the method of analysing them depends on the specifics. – AlphaNumeric Oct 25 '16 at 10:11
  • For the equation above, let $u=\log_{2}x$ then you have $u^{2}-u-2=0 \implies (u-2)(u+1)=0 \iff u= -1, u= 2$ hence solutions in $x$ are of the form $x=1/2, 4$. –  Oct 25 '16 at 10:33
  • What if we asume that $\log_2^2 x = \log_2 (log_2 x)$ – Nyklu Oct 25 '16 at 10:36
  • @Daniel Fisher In a vast majority of cases $\log_2^2x$ means $(\log_2 x)^2.$ The second possible meaning $\log_2 (\log_2 x)$ is only frequent in complexity issues, and usually written in the latter way to avoid ambiguities. – Jean Marie Oct 25 '16 at 10:40
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    Ok fair enough, well if you were to define it that way, then let $u=\log x$ for which the equation becomes $\log u - u=2$ hence, $\log u = u+2$ or $u=2^{2+u}$. Solutions to this type of equation require the use of the Lambert W Function. –  Oct 25 '16 at 10:58

1 Answers1

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Usually $\log^2$ is the same as $(\log)^2$.

This is because the $log$ function is a function, and you may denote it as $f$. Hence

$$f^2 = (f)^2$$

The other expression you wrote, namely $\log(\log)$ is not a square; this is a composition, namely (in terms of $f$) it's a $$f\circ f$$

And of course we have that in general

$$f\circ f \neq f^2$$

I have never seen the notation $\log^2$ used to mean $\log(\log)$. It would be messy and ill defined.

Enrico M.
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