Richter magnitude problem

Date: Wed, 23 Nov 1994 07:36:06 -0800 (PST)
From: Marla Friedman

Dr.Math-

I have been having some problem with the following question and many
others like it.  I was wondering if you could help me with it.

The Richter magnitude, R, of an earthquake is given by 

     R=0.67 log(0.37E)+1.46

where E is the energy in kW*h released by the earthquake.

--Show that if R increases by 1 unit, E increases by a factor of about 31.

I will appreciate any help you are willing to give me on how to solve
this problem.
                                Many Thanks,
                                        Marla Friedman

Date: 23 Nov 1994 16:50:26 GMT
From: Ken "Dr." Math
Organization: The Math Doctors

Hello there!

This is a problem that's basically about manipulating logarithms.  I'll
get you started, and then I'll let you take it from there.

What we want to do is start with the equation

R - 1 = 0.67 Log[0.37 E] + 1.46

And manipulate it until we get something of the form

R = 0.67 Log[0.37 X] + 1.46

The X we'll get will be in terms of E.  So let's get started.

R = 1 + 0.67 Log[0.37 E] + 1.46
R = 0.67(1.49254 + Log[0.37 E]) + 1.46

Now look at the part in the parentheses.  We want it to look like the Log
of some number, right?  Remember the formula Log[a] + Log[b] = Log[ab]? 
Well, we're going to use that, but first we have to get the 1.49254 into
Log form.  Can you figure out what the number a is such that Log[a] =
1.49254?  Once you've done that, you'll be almost done.

I'll let you take it from here.  Write back if you have more questions!

Ken "Dr." Math