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Isn't $\log x^m$ same as $m\cdot \log x$?

But the domain of let's say $\log x^2$ is negative infinity to positive infinity... on the other hand $2\log x$ is defined only for $x>0$

So given these two functions under what domain are these two functions identical?

What is wrong in the interpretation?

mrtaurho
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Rahul
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    They are the same wherever they are both defined. (also, you got the domain of $x\mapsto \log(x^2)$ wrong, as it should not include $0$). – Clement C. Oct 14 '18 at 17:42
  • We have $\log(x^2)=2\log x$ whenever both sides are defined. For negative $x$, only one of these expressions is regularly defined – Hagen von Eitzen Oct 14 '18 at 17:43
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    $\log x^m$ isn't the same as $m\cdot \log x$. It only works when $x$ is positive – Jakobian Oct 14 '18 at 17:44
  • $\log x^m$ and $m\log x$ are identical only when $x > 0$, since the second function is defined only when $x > 0$. Hence, the domain is $x \in \mathbb{R^+}$. – KM101 Oct 14 '18 at 18:13

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