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What am I doing wrong in this exponential equation?

  1. Dec 12, 2012 #1
    1. The problem statement, all variables and given/known data

    52x - 3(5x) = 10

    2. Relevant equations



    3. The attempt at a solution

    log 52x - log 3(5x) = log 10
    2x log 5 - log 3 - x log 5 = log 10
    x log 5 = log 10 + log 3
    x = (log 10 + log 3)/log 5
    putting this in my calculator I get 2.11
    by inspection I can tell that the answer will be x = 1. So what am I doing wrong?
     
  2. jcsd
  3. Dec 12, 2012 #2
    I think if you do the substitution [itex]u = 5^x[/itex] you can solve a quadratic for 'u' and then get 'x' from that. One of the 2 solutions will be negative and therefore outside of the domain of the log.

    Edit* Oh, well I guess I should say that the first line of your answer that you tried is already not right, you accidentally used the wrong properties for logs: http://dl.uncw.edu/digilib/mathematics/algebra/mat111hb/eandl/logprop/logprop.html
     
  4. Dec 12, 2012 #3

    SammyS

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    First of all, log(a + b) ≠ log(a) + log(b)

    To solve your equation, use the exponential property that, [itex]\displaystyle \ \ 5^{2x}=\left(5^x\right)^2\ .[/itex]

    Then letting u = 5x, you will have a quadratic equation in u .
     
  5. Dec 12, 2012 #4
    oh that makes more sense. I thought I could do that. Well thanks for the help I got the answer I needed.
     
  6. Dec 12, 2012 #5

    Ray Vickson

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    No, you cannot say 'log of a sum = sum of the logs'---that is just false. For example, log(5+1) = log(6) (because 5+1=6), but log(5) + log(1) = log(5) (because log(1) = 0).
     
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