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How do you find out the exponent if you can't use logs?

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



    2. Relevant equations



    3. The attempt at a solution

    There isn't a problem here, it's an example they gave, but I don't get how you can get the answer without using logs.

    2^(2x+1) = 1/32
    2^(2x+1) = 2^(-5)

    How do they know that it's -5?


    Same thing for:

    3^(x-1) = 27
    3^(x-1) = 3^3



    Is it just trial and error? I've tried a bunch of different methods but none have worked.
     
  2. jcsd
  3. Jul 12, 2012 #2

    Mark44

    Staff: Mentor

    Because 32 = 25, that's how. So 1/32 = 1/(25) = 2-5.
    You know that 32 = 9, right, so 33 = 27. That's all it is.
    No, it's not trial and error. It's useful to know a few powers of small numbers.

    2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 correspond to exponents on 2 of 1, 2, 3, ..., 10

    3, 9, 27, 81, 243 correspond to exponents on 3 of 1, 2, 3, 4, and 5

    and so on.
     
  4. Jul 12, 2012 #3
    Is there any simple formula I can use to find that 32 = 2^5? Or similar things, or is it just having an idea which numbers as the exponent would produce which numbers?

    Can I just knowing 32 = 2^x find out what x is with a formula?
     
  5. Jul 12, 2012 #4

    eumyang

    User Avatar
    Homework Helper

    Just memorize some of the powers, as Mark44 said. It really isn't much different than memorizing addition and multiplication facts (except that the numbers are bigger, of course).
     
  6. Jul 13, 2012 #5

    Mark44

    Staff: Mentor

    Or you might notice that 32 = 2 * 16 = 2 * 2 * 8 = 2 * 2 * 2 * 4 = 2 * 2 * 2 * 2 * 2 = 25.

    Do you know how to factor a number into its prime factors?
     
  7. Jul 13, 2012 #6
    Sure. You start out with 1, and then multiply it by 2 to get 2. Then you multiply the 2 by 2 to get 4. Then you multiply the 4 by 2 to get eight. Then you multiply the 8 by 2 to get 16. Then you multiply the 16 by 2 to get 32. How many multiplications did you do all together to get 32?
     
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