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Solving logs - Richter Scale and Decibels

  1. Jul 31, 2012 #1
    the text tells me that for calculating the Richter Scale magnitude of an earthquake we can use:

    M = log(I/I0) which can also be written as

    I = I0 x 10M

    Where M=magnitute, I=intensity, and I0=intensity 0


    How are those two formulas equal? Where did the log go? Can someone show me the proof for this?
     
  2. jcsd
  3. Jul 31, 2012 #2
    The log in the first equation must be to the base 10.

    To cancel the log you do:

    [itex]10^{M}=10^{Log(I/I_{0})}[/itex]

    Which becomes:

    [itex]10^{M}=I/I_{0}[/itex]

    Then multiply by [itex]I_{0}[/itex]

    [itex]I = 10^{M} * I_{0}[/itex]
     
  4. Jul 31, 2012 #3
    damn sorry the first one was supposed to be :

    M = 10log(I/I0)

    does that make a difference?
     
  5. Jul 31, 2012 #4

    Mark44

    Staff: Mentor

    That doesn't look right to me.

    Starting with your 2nd formula,

    I = I0 x 10M,
    Divide both sides by I0 to get I/I0 = 10M

    Now take the log (log10 or common log) of both sides.
     
  6. Jul 31, 2012 #5
    yes, I think I understand what to do once I have the formula I=I0x10M

    my problem is how to get to that from: M = 10log(I/I0)

    that is the formula they gave me.

    30 = 10log(I/I0)
    log(I/I0) = 3
    I = I0 x 103

    I have no idea how they figured out each step but that is what I was given and I want to know the proof for that.
     
  7. Jul 31, 2012 #6

    eumyang

    User Avatar
    Homework Helper

    Do you know the relationship between exponents and logarithms?
    log x = b iff 10b = x.

    So, looking at your last post, I'll insert a step.
    30 = 10log(I/I0)
    log(I/I0) = 3
    103 = I/I0
    I = I0 x 103

    Do you see it now?
     
  8. Jul 31, 2012 #7
    ooooooooh right wow.. how did I miss that? :P alright thanks!
     
  9. Jul 31, 2012 #8

    Mark44

    Staff: Mentor

    What I'm saying is that you can't get there from this formula. Here's why:
    M = 10log(I/I0)
    => M/10 = log(I/I0)
    => 10M/10 = I/I0
    => I = I010M/10

    This is different from the formula you show.

    Are you sure you're not misreading what they gave you? Or whoever wrote that formula might have made a typo, and typed "10log" instead of "log10".
     
  10. Aug 1, 2012 #9
    but thats exactly right? if you just sub in 30 for M it works out perfectly no?
     
  11. Aug 1, 2012 #10

    Mark44

    Staff: Mentor

    No, that's not exactly right. In post #9 I started with M = 10log(I/I0), solved for I, and got I = I010M/10.

    Your formula from post #1 is I = I010M.

    I hope that you can see that these are not the same.
     
  12. Aug 1, 2012 #11
    yeah I see what I did, it was just a mistake on the first post, but I get it now.

    also, just from looking at this question I was wondering:

    if M = log(I/I0) then I = I0 x 10M... would this be the same as... I = I0eM

    'e' as in exp.
     
  13. Aug 1, 2012 #12
    No. 10 does not equal e, does it?

    Logarithm to the base 10 is often denoted as "log"; logarithm to the base e is generally denoted as "ln".
     
  14. Aug 1, 2012 #13
    oh no sorry I meant 'E' as in scientific notation.
    like 99E7 = 99x107 = 990,000,000

    is that the same for the formula in my last post?
     
  15. Aug 1, 2012 #14
    Ah, yes. I've never seen it written that way before other than on a calculator, but I do know what you mean.
     
  16. Aug 1, 2012 #15

    Mark44

    Staff: Mentor

    They don't usually write the exponent as a superscript. With the E notation, it would be 99E7, or more likely, 9.9E8 or 9.9E08.
     
  17. Aug 1, 2012 #16
    ok right. But if M = log(I/I0) then I = I0 x 10M... would this be the same as... I = I0EM

    do you see what I'm saying? Because it would be a lot easier just to take the Io value and multiply it by 10 to the M every time, in such a situation.
     
  18. Aug 2, 2012 #17

    eumyang

    User Avatar
    Homework Helper

    You mean I = I0 x 10M.

    I guess so, but it's not usually written that way... only in calculators/computers. What if you have an exponential expression and the base is not 10?
     
  19. Aug 2, 2012 #18

    Mark44

    Staff: Mentor

    I doubt that anyone would look at I0EM and comprehend that M is supposed to be the exponent on 10. Instead, most people would interpret this as I0 * E * M, where E and M would be presumed to be some unstated values. I have never seen scientific notation in programming form (i.e., E+nn form) where the exponent is a variable.
     
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