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Math Help - IVT with ln Question

  1. #1
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    IVT with ln Question

    Hi there, I used to post here occasionally but my account was either deactivated or more likely I lost my password/username etc...

    Anyway, my question is this,

    use the intermediate value theorem to prove that equation lnx + x = 3 has a solution in the interval [1,e]

    Basically I know how to use the theorem but including ln and base e have lost me. Any help will be greatly appreciated! Thanks!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Swampscott, MA
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    Quote Originally Posted by Trevor View Post
    Hi there, I used to post here occasionally but my account was either deactivated or more likely I lost my password/username etc...

    Anyway, my question is this,

    use the intermediate value theorem to prove that equation lnx + x = 3 has a solution in the interval [1,e]

    Basically I know how to use the theorem but including ln and base e have lost me. Any help will be greatly appreciated! Thanks!
    You want to show that f(x)=\ln(x)+x-3 has a root in [1,e]. Evaluating f(1) gives us \ln(1)+1-3=-2. Evaluating f(e) gives us \ln(e)+e-3=e-2.

    Since f(1)<0 and f(e)>0, we know that (by the IVT) \exists~x_0\in[1,e] such that f(x_0)=0.
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