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Homework Help: Exp. Equation

  1. Jan 12, 2006 #1
    [tex]e^{2x + 1} = 5[/tex]

    How can I solve this without a calculator?
     
  2. jcsd
  3. Jan 12, 2006 #2

    EnumaElish

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    Do you need a number or an expression as an answer?
     
  4. Jan 12, 2006 #3
    For the other parts of the question I've been able to find a numerical answer so I assume I should be finding one for this one, but is it possible without a calculator?
     
  5. Jan 12, 2006 #4
    Not unless you can do ln(5) in your head or somehow...:uhh: Usually an expression is enough, depends on how it's being marked though.
     
  6. Jan 12, 2006 #5
    Well I get [tex]x=\frac{\ln(5)-1}{2}[/tex]. That doesn't have an exact decimal representation, so you can leave it in exact form or approximate.
     
  7. Jan 12, 2006 #6
    Ah ok, thanks.
     
  8. Jan 12, 2006 #7

    benorin

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    The method of solution is as follows:

    Given [tex]e^{2x + 1} = 5,[/tex]

    take the ln of both sides,

    [tex]\ln \left( e^{2x + 1}\right) = \ln (5),[/tex]

    recall that [itex]\ln \left( a^{x}\right) = x\ln \left( a\right) [/itex], so we have

    [tex](2x + 1)\ln \left( e\right) = \ln (5),[/tex]

    and since [itex]\ln \left( e\right) = 1 [/itex], we have

    [tex](2x + 1)(1) = \ln (5),[/tex]

    hence

    [tex]x=\frac{1}{2}\left( \ln (5) -1\right) [/tex]
     
  9. Jan 12, 2006 #8

    Hurkyl

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    Sure it does!

    Of course, I know you meant one that we can write with finitely many digits... but I don't want this to perpetuate the myth that decimal expansions do not exactly represent real numbers.
     
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