Differential: Richter Scale

On Richter scale, magnitude *M* of an earthquake of intesity *I* is:

*M* = (ln(I)) - ln(I)_{0})/(ln(10))

Where I_{0} is the min intensity used for comparison, and assume that it equals 1.

**a. Find intensity of 106 San Fran earthquake (M = 8.3)**

This is what I did:

I plugged in the givens.

8.3 = (ln I - ln(1))/ln(10)

I know that ln(1) = 0 so that gets taken out

I get left with:

8.3 = (ln I)/(ln 10)

Not sure if I did this correctly... But I multiplied both sides by ln 10.

But then I got lost when there was another ln on the other side.

I think I forgot something that was crucial in order to find the variable of something with a natural log.

Can someone remind me or help me out?

Thanks :)

__Answer: 10__^{8.3}=199,526,231.5

I think this is what they did:

I converted the last thing I did into exponential (the part where I multiplied both sides by 10)

__I=e__^{ln10*8.3}... Oh wait I just figured it out, it equals the answer, but it doesn't look like the 10^{8.3} thing xD. But yeah, if anyone wants to explain to me how

e^{ln10*8.3} = 10^{8.3} that would be nice :)

Also

**b. Find factor by which intensity is increased if Richter scale measurement is doubled**

I'm not really sure what to first do.

__The answer is: 10__^{R}

So I'm assuming it was the base to the power of R?

Is it really just like that, or was there more steps before it?

Thanks :)

Re: Differential: Richter Scale

So you want to solve ln(I)/ln(10)= 8.3 so that ln(I)= 8.3 ln(10).

Now, use the "laws of logarithms": [itex]ln(I)= ln(10^{83})[/itex]

If the Richter scale is doubled, that is 16.6 rather than 8.3, then you need to solve ln(I)/ln(10)= 16.6 so [itex]ln(I)= 16.6 ln(10)= ln(10^{16.6}[/itex]. Once you have found this I, divide by the previous one to find the factor.

Re: Differential: Richter Scale

Quote:

Originally Posted by

**HallsofIvy** So you want to solve ln(I)/ln(10)= 8.3 so that ln(I)= 8.3 ln(10).

Now, use the "laws of logarithms": [itex]ln(I)= ln(10^{83})[/itex]

If the Richter scale is doubled, that is 16.6 rather than 8.3, then you need to solve ln(I)/ln(10)= 16.6 so [itex]ln(I)= 16.6 ln(10)= ln(10^{16.6}[/itex]. Once you have found this I, divide by the previous one to find the factor.

Oh right!

I forgot!

When there's a number in front of a log, which is the coefficient, then it becomes the exponent!

But then, you would just divide __ln__ from both sides right? If that's possible.

So yeah, thanks! :)

And I assume you do the same thing for letter b.?

Have it equal to 16.6 instead of 8.3

Then go through the same process?

By the way, it's actually 10^{M}, my bad, accidently put R.

But wait, the answer has the M in it, so would I plug anything in?