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Solving for x in exponent

  1. Oct 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Solve the equation [itex] 4^{x-1} = 2^x + 8[/itex].

    2. Relevant equations
    Just algebra

    3. The attempt at a solution
    [itex] 4^{x-1} = 2^x + 8[/itex]
    [itex] 2^{2(x-1)} = 2^x + 2^3[/itex]
    [itex] 2^{2x}2^{-2} = 2^x + 2^3 [/itex]
    [itex] \frac{2^{2x}}{2^2} = 2^x + 2^3 [/itex]
    [itex] 2^{2x} = 2^22^x + 2^5 [/itex]
    [itex] 2^x2^x - 2^22^x = 2^5 [/itex]
    [itex] 2^x(2^x - 2^2) = 2^5 [/itex]

    I made a few jumps here and there in my work, but it should make sense. This is about as far as I get before I get stuck. My intuition is to try to get this down to two terms with the same base, 2, so I can just equate their powers, but I can't seem to get rid of any of the terms, so I'm not sure where to go from there. I'm sure I'm missing something obvious, but I can't see it at the moment. Do you have any tips on what a better angle to look at this problem from is? Thanks!
     
  2. jcsd
  3. Oct 23, 2015 #2

    BvU

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    Try to think this coming from the other end: what kinds of equations are you supposed to be able to solve ? linear, quadratic, goniometric. So perhaps one of these is carefully hidden in this exercise -- and the quadratic kind is the most likely candidate.
    When you then realize that ##2^{2x} = (2^x)^2## and look at your line 3 with that knowledge ...
     
  4. Oct 23, 2015 #3

    Mark44

    Staff: Mentor

    ##4^{x - 1} = 4^x \cdot 4^{-1} = \frac 1 4 4^x##
    Move all terms over to the left side, and multiply both sides by 4.
    The equation is quadratic in form, and can be factored, as BvU suggests.
     
    Last edited by a moderator: Oct 24, 2015
  5. Oct 23, 2015 #4
    Thanks for your help! I figured it out after realizing it made a quadratic and used the quadratic formula.
     
  6. Oct 24, 2015 #5

    SammyS

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    Yes. the next to last equation in your OP
    ##\displaystyle \ 2^x2^x - 2^22^x = 2^5 \ ##​
    can be written as
    ##\displaystyle \ \left(2^x\right)^2 - 4\cdot2^x = 32 \ ## .​
     
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