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Thread: 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 $\displaystyle f(x)=\ln(x)+x-3$ has a root in $\displaystyle [1,e]$. Evaluating $\displaystyle f(1)$ gives us $\displaystyle \ln(1)+1-3=-2$. Evaluating $\displaystyle f(e)$ gives us $\displaystyle \ln(e)+e-3=e-2$.

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