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Math Help - rewrite exponential function

  1. #1
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    rewrite exponential function

    y=2^x

    rewrite in terms of e
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by ToastE View Post
    y=2^x

    rewrite in terms of e
    Let X=2^x

    We know that e^{\ln(X)}=X

    --> 2^x=e^{\ln(2^x)}

    Using the rule \ln(a^b)=b \ln(a) :

    2^x=e^{x \cdot \ln(2)}


    Thus y=2^x \Longleftrightarrow y=e^{x \cdot \ln(2)}







    Late edit : ok, that was my 3th post ! xD
    Last edited by Moo; September 5th 2008 at 10:44 PM.
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  3. #3
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    Quote Originally Posted by ToastE View Post
    y=2^x

    rewrite in terms of e
    Note that:

    a^x = \left(e^{\ln a}\right)^x = e^{x \ln a}
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  4. #4
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    Quote Originally Posted by Moo View Post
    Hello,

    Let X=2^x

    We know that e^{\ln(X)}=X

    --> 2^x=e^{\ln(2^x)}

    Using the rule \ln(a^b)=b \ln(a) :

    2^x=e^{x \cdot \ln(2)}


    Thus y=2^x \Longleftrightarrow y=e^{x \cdot \ln(2)}
    The only thing I am confused about is the math]e^{\ln(X)}=X[/tex]
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  5. #5
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    THANK YOU
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  6. #6
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by ToastE View Post
    The only thing I am confused about is the math]e^{\ln(X)}=X[/tex]

    Taking logarithms is the inverse of raising to a power. So a natural logarithm is the inverse of raising to power e. It just is, that's how it's defined.

    So x = \ln y means the same as y = e^x.

    what you're saying when you say x = \ln y is: "What number do I raise e to the power of in order to get y?"

    So e^{\ln(X)}=X means "e to the power of the number you have to raise e to the power of to get X is X."
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  7. #7
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    To demonstrate what Matt said:

    \ln{a} = b

    a = e^b

    \therefore a = e^{\ln{a}}
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  8. #8
    Super Member Matt Westwood's Avatar
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    In case you wonder, \ln \left({e^X}\right) = X as well.
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