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A petite observation about logarithms

  1. Nov 15, 2005 #1
    Here's something cute:

    Consider the graph of ln(x^2) and then consider the graph of 2ln(x), missing anything?

    I was momentarily caught off guard by this until I realized that when we derive the property: ln(x^a)=aln(x), we choose the positive root.

    Has anyone ever run into a situation where it was better to say that ln(x^2)=2ln|x| ?
  2. jcsd
  3. Nov 15, 2005 #2
    Well, consider the case x < 0. Then ln(x^2) is real, but 2ln(x) has a non-zero imaginary part, so clearly they can't be equal.
  4. Nov 15, 2005 #3


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    Yes, that's a very good point.

    Solving the equation ln(x^2)= 0 is NOT the same as solving 2ln(x)= 0 and, yes, it is better to write ln(x^2)= 2ln(|x|).
  5. Nov 20, 2005 #4

    D H

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    On most computers it is not better to write ln(x^2) instead of 2ln(|x|) in a computer program.
  6. Nov 20, 2005 #5


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    I'm sorry, what does that have to do with my response? My point was NOT that it was better to write ln(x^2) rather than 2ln(|x|) but rather that it was better to write ln(x^2)= 2ln(|x|) rather than ln(x^2)= 2ln(x).
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