## Solving logs - Richter Scale and Decibels

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?
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 The log in the first equation must be to the base 10. To cancel the log you do: $10^{M}=10^{Log(I/I_{0})}$ Which becomes: $10^{M}=I/I_{0}$ Then multiply by $I_{0}$ $I = 10^{M} * I_{0}$
 damn sorry the first one was supposed to be : M = 10log(I/I0) does that make a difference?

Mentor

## Solving logs - Richter Scale and Decibels

 Quote by Gregory.gags damn sorry the first one was supposed to be : M = 10log(I/I0)
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.
 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.
 Recognitions: Homework Help 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?
 ooooooooh right wow.. how did I miss that? :P alright thanks!

Mentor
 Quote by Gregory.gags 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)
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".
 Quote by Gregory.gags 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.

 Quote by Mark44 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".
but thats exactly right? if you just sub in 30 for M it works out perfectly no?
 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.
 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.

 Quote by Gregory.gags if M = log(I/I0) then I = I0 x 10M... would this be the same as... I = I0eM 'e' as in exp.
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".

 Quote by oay 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".
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?

 Quote by Gregory.gags 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?
Ah, yes. I've never seen it written that way before other than on a calculator, but I do know what you mean.

Mentor
 Quote by Gregory.gags 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?
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.
 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.

Recognitions:
Homework Help
 Quote by Gregory.gags ok right. But if M = log(I/I0) then I = I0 x 10M...
You mean I = I0 x 10M.

 Quote by Gregory.gags ...would this be the same as... I = I0EM
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?

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