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Why this property is always written without the module?

I always find $\log x^k = k \log x$, but IMO the correct form (when $k$ is even) is $\log x^k = k\log |x|$.

E.g. $\log(-100)^2 = 2 \log |-100| = 4\log10$ and not equal to $2 \log (-100)$ which is a complex number.

Am I wrong?

Start wearing purple
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collimarco
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    Necessity for the absolute value depends on what $x$ is. – Git Gud Aug 25 '13 at 10:40
  • no, you're right. But perhaps it's just that $x$ is always assumed to be positive? Though, when you're solving equations, that incomplete rule might leave you without certain roots ;) – W_D Aug 25 '13 at 10:40
  • For the same reason people often say 'Good morning' instead of 'I wish you have a good morning'. What is missing is being assumed from the context. – OR. Aug 25 '13 at 11:26

2 Answers2

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Your expression is correct. It is also correct for $k$ a rational number of the form $a/b$, where $a$ and $b$ are integers with $a$ even and $b$ odd.

However, when we are going to use logarithms, it is best if we arrange things so that we avoid expressions of the form $c^d$, where $c$ is or could be negative.

André Nicolas
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Because in most cases we assume from the beginning that $\,x>0\,$ and thus the absolute value is superfluous.

Yet this is not the case if you have an even power of a negative real number...

DonAntonio
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