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Math Help - Proving ln( 5 root e) = 1/5

  1. #1
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    Proving ln( 5 root e) = 1/5

    I need help with this proof...

    Prove that ln 5 sqrt{e} = 1/5

    It is actually like ln e^(1/5) but I dont know how to make the 5 small...
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  2. #2
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    Quote Originally Posted by circuscircus View Post
    I need help with this proof...

    Prove that ln 5 sqrt{e} = 1/5

    It is actually like ln e^(1/5) but I dont know how to make the 5 small...
    You mean ln[fifth root of e].
    "...small 5 sqrt...." ----------that is fifth root.

    So, prove ln[e^(1/5)] = 1/5

    We will use:
    ---Log[a^b] = b*Log[a]
    ---Log(base a)[a] = 1

    ln[e^(1/5)]
    = (1/5)ln[e]
    = (1/5)*Log(base e)[e]
    = (1/5)*1
    = 1/5
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  3. #3
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    Where does this come from?

    Log[a^b] = b*Log[a]
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  4. #4
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    Quote Originally Posted by circuscircus View Post
    Where does this come from?

    Log[a^b] = b*Log[a]
    Recall that
    log(n \cdot a) = log(n) + log(a)

    What does a^b mean? It means a^b = a\cdot a \cdot ~ ... ~ \cdot a where we are doing this "b" times.

    So
    log(a^b) = log(a) + log(a) + ~ ... ~ + log(a) = b \cdot log(a)

    Note that it does not matter what base the logarithm is to:
    log_{10}(a^b) = b \cdot log_{10}(a)
    and
    log_2(a^b) = b \cdot log_2(a)

    -Dan
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