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Math Help - exp and log

  1. #1
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    exp and log

    He,

    I want to prove that exp(log(x))=log(exp(x))=x

    Now the second one i diff. and get 1/exp(x) * exp(x) = 1
    So log(exp(x))= x+ C, Calculation shows c = 0
    But hopw do I get the first one

    because when i diff. I get exp(log(x))/x , now I have to prove that exp(log(x))=x and so this doesn't help

    What can I do now?

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  2. #2
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    Quote Originally Posted by kuntah View Post
    He,

    I want to prove that exp(log(x))=log(exp(x))=x

    There is nothing to prove, you just need the definition of the natural log.

    If y=\log(x), then \exp(y)=x, so by the definition:

    \exp(y)=\exp(\log(x))=x

    (this is taking \exp of y=\log(x))

    Similarly, if x=\log(y), then \exp(x)=y, so taking \log of this last expression we have:

    \log(\exp(x))=\log(y)=x

    RonL
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  3. #3
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    yes thanks but,

    The problem I had to solve was to prove exp(log(x)=x
    So I don't know actually yet, according to the exercise, that this is true
    But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

    Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
    This must be equal to 1 then. Then exp(log(x)) must be x, because then
    exp(log(x))/x=1, but I can't use this, because this is what I have to prove
    Can't I use something else?

    Thanks
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kuntah View Post
    yes thanks but,

    The problem I had to solve was to prove exp(log(x)=x
    So I don't know actually yet, according to the exercise, that this is true
    But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

    Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
    This must be equal to 1 then. Then exp(log(x)) must be x, because then
    exp(log(x))/x=1, but I can't use this, because this is what I have to prove
    Can't I use something else?

    Thanks
    what does the relationship in question remind you of? it reminds me of the relationship between inverse functions. recall that, a function f(x) is invertible if there exists a function f^{-1}(x) such that:

    f \left( f^{-1}(x) \right) = f^{-1}(f(x)) = x

    so answering your question amounts to proving that the log is the inverse of the exponential function. can you do that?
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  5. #5
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    Quote Originally Posted by kuntah View Post
    yes thanks but,

    The problem I had to solve was to prove exp(log(x)=x
    So I don't know actually yet, according to the exercise, that this is true
    But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

    Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
    This must be equal to 1 then. Then exp(log(x)) must be x, because then
    exp(log(x))/x=1, but I can't use this, because this is what I have to prove
    Can't I use something else?

    Thanks
    But you do know:

    "If , then ",

    and:

    "if , then "

    which are all I used to show that:



    \exp(\log(x))=\log(\exp(x))=x

    No calculus needed, just the definition of natural log.

    RonL
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  6. #6
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    Quote Originally Posted by kuntah View Post
    I want to prove that exp(log(x))=log(exp(x))=x
    I would like to know more about the background of this question. There is a widely use calculus textbook by Salas and Hille that asks this very question. That text defines the logarithm function as: Log(x) = \int\limits_1^x {\frac{{dt}}{t}} ,\quad x > 0.
    Further, they define e as the number such that Log(e) = \int\limits_1^e {\frac{{dt}}{t}}  = 1.

    From these, the authors develop the usual properties of both Log(x) and exp(x).
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  7. #7
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    I defined exp as the solution to this differential equation

    x'(t) = f(x(t)) with boundary values (0,1)

    for the log I used the same definition
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  8. #8
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    Quote Originally Posted by kuntah View Post
    I defined exp as the solution to this differential equation

    x'(t) = f(x(t)) with boundary values (0,1)

    for the log I used the same definition
    You mean x'(t)=x(t) with initial value x(0)=1

    RonL
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  9. #9
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    Quote Originally Posted by kuntah View Post
    He,

    I want to prove that exp(log(x))=log(exp(x))=x

    Now the second one i diff. and get 1/exp(x) * exp(x) = 1
    So log(exp(x))= x+ C, Calculation shows c = 0
    But hopw do I get the first one

    because when i diff. I get exp(log(x))/x , now I have to prove that exp(log(x))=x and so this doesn't help

    What can I do now?

    How about

    e^(ln(x)) = N
    Take the natural logs of both sides,
    ln(x) *ln(e) = ln(N)
    ln(x) = ln(N)
    N = x
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