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Logarithm rearrangement

  1. Mar 21, 2017 #1
    1. The problem statement, all variables and given/known data
    2 - log10 3x = log10(x/12)

    2. Relevant equations
    logab=b log a
    log(a/b)= log a - log b

    3. The attempt at a solution
    2 + log10 12= log10 x - x log10 3
    Start seems simple but cannot see where to go from here, taking exponentials doesn't seem to help. Not sure what the next steps could be.
     
  2. jcsd
  3. Mar 21, 2017 #2
    log is base 10 or it is natural log ?
     
  4. Mar 21, 2017 #3
    I assume it is meant to be base 10, so I have edited post to include the base.
     
  5. Mar 21, 2017 #4
    After rearranging a bit I can't see how this has a nice solution.

    I get ##5^2 *2^4*3 = x3^x##.
     
  6. Mar 21, 2017 #5
    I'm happy with that as well, thanks.
     
  7. Mar 21, 2017 #6

    Mark44

    Staff: Mentor

    Your equation is equivalent to ##100 = \frac x {12} \cdot 3^x##. Because the variable occurs both as an exponent and as a multiplier, there are not any simple analytic ways to solve this equation. However, you can get good approximations by numeric means, simply by substituting value for x on the right side, and comparing the result with 100 on the left side. Using a spreadsheet I see that there is a solution near x = 50.16. The actual solution is slightly smaller than this.

    Edit: As Ray points out, my number here is incorrect. I was using an incorrect formula in my spreadsheet, using log(3^x) instead of 3^x.
     
    Last edited: Mar 21, 2017
  8. Mar 21, 2017 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    For If ##f(x) = (x/12) 3^x,## we have##f(50) = (50/3) e^{50} \doteq 0.299 \times 10^{25}##, so the solution of ##f(x) = 100## must certainly be quite a bit less than 50. Maple gets ##x \doteq 4.990.##
     
  9. Mar 21, 2017 #8
    what is ##\doteq## ?
     
  10. Mar 21, 2017 #9

    Ray Vickson

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    Science Advisor
    Homework Helper

    ##\doteq## means "approximately equal to", sometimes also written as ##\approx##. I avoid using "=" in such cases just so the reader will understand that the answer is not exactly 4.990. For example, a better approximation is obtained by using 60 digits of precision, giving
    ##x \doteq 4.99043541467729841484302401855197675632523233638262678465047## Even that is not exact.
     
  11. Mar 21, 2017 #10
    As a additional exercise, Can we prove that there is no nice real solution for equation, Also can we know the nature of the solution ?
     
  12. Mar 21, 2017 #11

    Mark44

    Staff: Mentor

    You are correct. Somehow I mistakenly had log(3^x) in my spreadsheet formula, not log(3^x) as it should have been.
     
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