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Math Help - Logarithmic exponenent

  1. #1
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    Logarithmic exponenent

    How would I read an equation like this 6^(log_6 20)=20 ? How would I evaluate it? Why is it equal to 20?
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    MHF Contributor Siron's Avatar
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    Re: Logarithmic exponenent

    In general y=\ln(x) \Leftrightarrow e^{y}=x \Leftrightarrow e^{\ln(x)}=x. Do you recognize this in your example?
    You can also take the \log_6 of both sides and simplify with the logarithm rules.
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    Re: Logarithmic exponenent

    I am sorry, but I do not quite understand the defintion you provided. What do the arrows represent?
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    MHF Contributor Siron's Avatar
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    Re: Logarithmic exponenent

    The arrows indicate that it's equivalent to write y=\ln(x) or e^{y}=x for example. But do you understand it now?
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    Re: Logarithmic exponenent

    I think I am beginning to. Why is it, though, that when the base of the logarithm in the exponent and the base of that logarithmic exponenent are equal, the answer is arguement? Is it because it is not an algebraic function, and is read differently because of that.
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Logarithmic exponenent

    It's just a very handy proposition to work with logarithms. You'll see if you take the \log_6 of both sides then you get:
    \log_6(6^{\log_6(20)})=\log_6(20)
    \Leftrightarrow \log_6(20)\cdot \log_6(6)=\log_6(20)
    \Leftrightarrow \log_6(6)=1
    \Leftrightarrow 6^1=6

    So indeed, they LHS and the RHS of the expression are equal.
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