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Math Help - Logarithimic/Exponential equations

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    Logarithimic/Exponential equations

    http://img3.imageshack.us/img3/9558/img163e.jpg

    1.2, 1.4, and 1.5

    I need big help on 1.4 and 1.5.

    I think x = 5 for 1.2. I think 1.4 and 1.5 are factorable, but I don't know how to go about doing that. No examples in my textbook...
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    Quote Originally Posted by A Beautiful Mind View Post
    http://img3.imageshack.us/img3/9558/img163e.jpg

    1.2, 1.4, and 1.5

    I need big help on 1.4 and 1.5.

    I think x = 5 for 1.2. I think 1.4 and 1.5 are factorable, but I don't know how to go about doing that. No examples in my textbook...
    1.2 has no solution. There is no number that makes e^{x-5} = 0

    ============================================

    1.4 e^{-x}-3e^x+2=0

    Multiply by e^x : 1-3e^{2x}+2e^x = 0

    3e^{2x}-2e^x-1= (3e^x+1)(e^x-1) = 0

    The first solution will be complex and the second real

    x = ln\left(\frac{1}{3}\right) + i\pi or x = 0

    ===========================================

    1.5: e^{5x}-2e^{3x}-3e^x=0

    Divide by e^x (see 1.2 as to why e^x=0 is not a solution)

    e^{4x}-2e^{2x}-3=0

    Set u=e^{2x} \: \rightarrow \: u^2-2u-3=0

    (u-3)(u+1) = 0

    e^{2x} = 3 or e^{2x} = -1

    2x = ln(3) \: \rightarrow \: x = \frac{1}{2}ln(3)

    OR

    2x = i\pi \: \rightarrow \: x = \frac{1}{2}i\pi
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