Algebra problem (earthquake magnitude)?

Question

The 2004 earthquake in Southern California had a magnitude of 6.2 on the Richter scale. The 1903 earthquake was 9.5 on the Richter scale. Find the RATIO of the intensities of the earthquake.

Note: The magnitude R, measured on the Richter scale, of an earthquake intensity I is defined as R = log I/I(base 0), where I(base 0) is a minimum intensity for comparison.

Can anyone please show me how to do this?

Answer 1

First, your quesion has an error, because "0" is not a correct value for a logrithm base. The purpose of logirithm is to determine the value for x in the relationship y=n^x, when y and n are known. 'n' is the base of the logrithm and n=0 would not provide a very useful logrithm since zero to any non-zero power is zero.

Also, when y1 = n^x1 and y2 = n^x2 then y1*y2 = (n^x1)*(n^x2)

This is simplified to y1*y2 = n^(x1 + x2).
Therefore, the log(y1*y2)= x1 + x2 = log(y1) + log(y2)

Similarly, y1/y2 = (n^x1)/(n^x2) and simplifies to y1/y2=n^(x1-x2).
Therefore log(y1/y2)= x1 - x2 = log(y1) - log(y2)

Since you seem to have a mistake in your equation, I looked up the relationship between energy content of an earthquake and Richter scale magnitude. I found that it is:

2/3*log(Energy)+constant = Ricther magnitude.

For Earthquake 2004
2/3*log(E)+constant=6.2
log(E)=(6.2-constant)*3/2
E_2004=10^(3/2*(6.2-constant))

For Earthquake 1903:
E_1903=10^(3/2*(9.5-constant))

E_1903/E_2004 = 10^(3/2*(9.5-constant)) / 10^(3/2*(6.2 - constant))

E_1903/E_2004 = 10^(3/2*(9.5-constant)-3/2*(6.2-constant))
E_1903/E_2004 = 10^(3/2*(9.5-6.2))
E_1903/E_2004 = 10^(3/2*(3.3)) = 10^4.95 = 89125

You may need to change the math shown above when you substitute the correct equation for the relationship between Ricther magnitude and Intensity.

http://www.ux1.eiu.edu/~cfjps/1300/magnitude.html

Answer 2

The ratio would be:

10 to the power of (9.5 - 6.2)
since the inverse of log() is 10^x

10^3.3 = 1995 approximately 2000

The 1903 earthquatke was about 2000 times greater than the 2004 quake.

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