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Math Help - Logarithmic Equations

  1. #1
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    Logarithmic Equations

    1) ln(1/sqrt(1+x^2)) +1=0
    ???
    2) e^x=1+2e^-x

    for this one I've got e^(x)-2e=1, then e(x-2)=1. I'm just unsure of where to go from here?
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  2. #2
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    Quote Originally Posted by Chinnie15 View Post
    1) ln(1/sqrt(1+x^2)) +1=0
    ???
    2) e^x=1+2e^-x

    for this one I've got e^(x)-2e=1, then e(x-2)=1. I'm just unsure of where to go from here?
    1) \ln\left(\frac{1}{\sqrt{1+x^2}} \right) + 1 = 0

    \ln\left(\frac{1}{\sqrt{1+x^2}} \right)  = -1

    change to an exponential equation ...

    e^{-1} = \frac{1}{\sqrt{1+x^2}}

    solve for x

    2) e^x = 1 + 2e^{-x}

    multiply every term by e^x ...

    e^{2x} = e^x + 2

    e^{2x} - e^x - 2 = 0

    factor the quadratic and solve for x
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  3. #3
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    how come you multiply every term by e^x? I don't see that.
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  4. #4
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    Quote Originally Posted by Chinnie15 View Post
    how come you multiply every term by e^x? I don't see that.
    well, now you've seen it ...

    I wanted to get rid of the e^{-x} in the equation.

    it's like if you had the equation y = 1 + \frac{2}{y} ... you'd multiply every term by y to clear the unwanted fraction.

    ... and it makes the equation quadratic in e^x, which can be easily factored and solved.
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  5. #5
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    ok, I see it now. thank you!
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