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I need sums in LOG

  1. Feb 22, 2006 #1
    Hi,

    I just need a little help in getting some sums. Can anyone of you give me a site where I can find sums in Log so that I can do them and practice a lot.

    I mean like sums in this type

    Show that log(xy)base16 = 1/2log(X)base4 + 1/2log(Y)base4

    Thanks just need some sums of this type to practice my self.

    Thanks alot people just give me a few links:smile:

    Thanks
     
  2. jcsd
  3. Feb 22, 2006 #2

    arildno

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    Okay, don't bother with the base change first:
    Firstly:
    How can you change your left-hand side from a product into a sum?
     
  4. Feb 23, 2006 #3
    Well I mean

    Ok I mean not converting to a sum. I mean to prove that you can convert it to a sum.
    I mean to prove only 1 side to get the left hand side. And then show that it could be proved.

    I think I expressed in the correct way because I am from a non-english country now learning in the english medium
     
  5. Feb 23, 2006 #4

    arildno

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    Well, but a fundamental property about any log is that we have log(xy)=log(x)+log(y)
     
  6. Feb 23, 2006 #5
    Ya ya I know that, but you can convert it to sums like I've shown above isn't it?
     
  7. Feb 23, 2006 #6

    arildno

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    Let's take it in detail.
    We have:
    [tex]\log_{16}(xy)=\log_{16}(x)+\log_{16}(y)[/tex]
    by the fundamental property of logs.

    Now, we need to relate logs with different bases!
    We have, for bases a, b, the identities:
    [tex]x=a^{\log_{a}(x)}=b^{\log_{b}(x)}, a=b^{log_{b}(a)[/tex]
    Thus, we get:
    [tex]b^{\log_{b}(x)}=(b^{\log_{b}(a)})^{\log_{a}(x)}=b^{\log_{b}(a)\log_{a}(x)}[/tex]
    Since logs are unique, we therefore have:
    [tex]\log_{b}(x)=\log_{b}(a)\log_{a}(x)[/tex]
    Now, let b=4, a=16, and get your result.
     
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