# Homework Help: Solving logs - Richter Scale and Decibels

1. Jul 31, 2012

### Gregory.gags

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. Jul 31, 2012

### rollcast

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 $I_{0}$

$I = 10^{M} * I_{0}$

3. Jul 31, 2012

### Gregory.gags

damn sorry the first one was supposed to be :

M = 10log(I/I0)

does that make a difference?

4. Jul 31, 2012

### Staff: Mentor

That doesn't look right to me.

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.

5. Jul 31, 2012

### 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)

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.

6. Jul 31, 2012

### eumyang

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?

7. Jul 31, 2012

### Gregory.gags

ooooooooh right wow.. how did I miss that? :P alright thanks!

8. Jul 31, 2012

### 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".

9. Aug 1, 2012

### Gregory.gags

but thats exactly right? if you just sub in 30 for M it works out perfectly no?

10. Aug 1, 2012

### 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.

11. Aug 1, 2012

### Gregory.gags

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.

12. Aug 1, 2012

### skiller

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".

13. Aug 1, 2012

### 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?

14. Aug 1, 2012

### skiller

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

15. Aug 1, 2012

### 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.

16. Aug 1, 2012

### Gregory.gags

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.

17. Aug 2, 2012

### eumyang

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?

18. Aug 2, 2012

### 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.