Solving logs - Richter Scale and Decibels

[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 10^M

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?

 

[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}

 

[Gregory.gags]

damn sorry the first one was supposed to be :

M = 10log(I/I₀)

does that make a difference?

 

[Mark44]

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

Now take the log (log₁₀ or common log) of both sides.

 

[Gregory.gags]

yes, I think I understand what to do once I have the formula I=I₀x10^M

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.

 

[eumyang]

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

 

[Gregory.gags]

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

 

[Mark44]

yes, I think I understand what to do once I have the formula I=I₀x10^M

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)
=> 10^M/10 = I/I₀
=> I = I₀10^M/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".

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.

 

[Gregory.gags]

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)
=> 10^M/10 = I/I₀
=> I = I₀10^M/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 that's exactly right? if you just sub in 30 for M it works out perfectly no?

 

[Mark44]

No, that's not exactly right. In post #9 I started with M = 10log(I/I₀), solved for I, and got I = I₀10^M/10.

Your formula from post #1 is I = I₀10^M.

I hope that you can see that these are not the same.

 

[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 10^M... would this be the same as... I = I₀e^M

'e' as in exp.

 

[skiller]

if M = log(I/I₀) then I = I₀ x 10^M... would this be the same as... I = I₀e^M

'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".

 

[Gregory.gags]

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 99_E⁷ = 99x10⁷ = 990,000,000

is that the same for the formula in my last post?

 

[skiller]

oh no sorry I meant '_E' as in scientific notation.
like 99_E⁷ = 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.

 

[Mark44]

oh no sorry I meant '_E' as in scientific notation.
like 99_E⁷ = 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.

 

[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.

 

[eumyang]

ok right. But if M = log(I/I₀) then I = I₀ x 10M...

You mean I = I₀ x 10^M.

...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?

 

[Mark44]

ok right. But if M = log(I/I₀) then I = I₀ x 10M... would this be the same as... I = I₀EM

You mean I = I₀ x 10^M.

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.

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?

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.