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Math Help - ln(1/a) = -ln(a)

  1. #1
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    ln(1/a) = -ln(a)

    how can we prove ln(1/a) = -ln(a)

    by using fundermental theory of calculus?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ShaXar View Post
    how can we prove ln(1/a) = -ln(a)

    by using fundermental theory of calculus?
    Look at the definition of \ln (presumably you have a definition using calculus) then derive the laws of logarithms and apply them

    CB
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  3. #3
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    Quote Originally Posted by ShaXar View Post
    how can we prove ln(1/a) = -ln(a)

    by using fundermental theory of calculus?

    Let a > 0, then \ln a = \int\limits^a_1 \frac{dt}{t}; make change of variable: u = \frac{1}{t}\Longrightarrow du = -\frac{dt}{t^2}\Longrightarrow dt = -\frac{du}{u^2} , and the limits: t = 1 \Longrightarrow u = 1\,,\,t=a\Longrightarrow u=\frac{1}{a} , so :

    \ln a = \int\limits^a_1 \frac{dt}{t}=-\int\limits_1^{1/a}\frac{du}{u}=-\ln(1/a)

    Tonio
    Last edited by tonio; May 7th 2010 at 04:23 AM. Reason: Correcting little typo
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  4. #4
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    Quote Originally Posted by ShaXar View Post
    how can we prove ln(1/a) = -ln(a)

    by using fundermental theory of calculus?
    Me too it is a similar problem that brought me to this site. Essentially ShaXar wants to find the intersection of the curves exp(x) and 1/x or the curves 1/x and ln(x), i.e. where they intersect but it is not so simple.

    How on earth can one solve such an ordinary and seemingly so simple a problem, anyone ???
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  5. #5
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    Quote Originally Posted by Khalfan View Post
    Me too it is a similar problem that brought me to this site. Essentially ShaXar wants to find the intersection of the curves exp(x) and 1/x or the curves 1/x and ln(x), i.e. where they intersect but it is not so simple.

    How on earth can one solve such an ordinary and seemingly so simple a problem, anyone ???
    If e^x= \frac{1}{x}, then xe^x= 1 so x= W(1) where W(x) is the "Lambert W function", Lambert W function - Wikipedia, the free encyclopedia.
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Khalfan View Post
    Me too it is a similar problem that brought me to this site. Essentially ShaXar wants to find the intersection of the curves exp(x) and 1/x or the curves 1/x and ln(x), i.e. where they intersect but it is not so simple.

    How on earth can one solve such an ordinary and seemingly so simple a problem, anyone ???
    No he does not, what he needs to do is read tonio's post (not that I would be so explicit in explaining it myself obviously).

    CB
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