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Math Help - Proving Law of Exp and Log

  1. #1
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    Proving Law of Exp and Log

    1. Given the product law of logarithms, prove the product law of exponents.

    2. Given the quotient law of logarithms, prove the quotient law of exponents.

    3. Apply algebraic reasoning to show that
    a=b^(loga/logb) for any a,b>0

    Please explain these to me.

    All I know is that

    The product law of logs are:

    Log(AB)=logA+logB

    The Quotient law of logs are:

    Log(A/B)=logA-Logb

    Edit:



    For product law:
    let m=b^x and let n=b^y
    mn=(b^x )(b^y)
    and then what should i do?
    Last edited by skeske1234; May 23rd 2009 at 03:13 PM.
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    ...

    3. Apply algebraic reasoning to show that
    a=b^(loga/logb) for any a,b>0

    Please explain these to me.

    ...
    I assume that loga means \log_{10}(a). If so:

    You are supposed to know: a = 10^{\log_{10}(a)}

    a=b^{\dfrac{\log_{10}(a)}{\log_{10}(b)}}

    a^{\log_{10}(b)}=\left( b^{\dfrac{\log_{10}(a)}{\log_{10}(b)}} \right)^{\log_{10}(b)}

    a^{\log_{10}(b)}= b^{\log_{10}(a)}

    Now express the values of a and b as a power to the base 10:

    \left(10^{\log_{10}(a)}  \right)^{\log_{10}(b)} = \left(10^{\log_{10}(b)}  \right)^{\log_{10}(a)}~\implies~\boxed{10^{{\log_{  10}(a)}  \cdot {\log_{10}(b)}} = 10^{{\log_{10}(b)}  \cdot {\log_{10}(a)}}}
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