The magnitude M on the Richter scale of an earthquake as a function of its inten
ID: 2972558 • Letter: T
Question
The magnitude M on the Richter scale of an earthquake as a function of its intensity I is given by
M= log base 10 (I / Io)
where Io is some fixed reference level of intensity.
The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. Suppose that at the same time in South America there was an earthquake with magnitude 5.2 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one?
Answer-?
Hint: You don't need to know what Io is. Plug in the respective magnitudes and use the log properties you know to compare the resulting values for I in terms of this Io .
Explanation / Answer
Now, M=log(I/Io). Since the earthquake in San Francisco had a magnitude of 8.3, M=8.3 in this case. The earthquake in South America had a magnitude of 5.2, so M=5.2 in that case. 8.3=log(I/Io), so by definition of the logarithm, this is the exponent you must raise the base 10 by in order to obtain I/Io. Thus, 10^8.3=I/Io, or Io*10^8.3=I. Analogously, 5.2=log(I/Io), so 10^5.2=I/Io, or Io*10^5.2=I. Take their ratio, putting the value of I for the San Francisco earthquake in the numerator: Io*10^8.3/(Io*10^5.2)=10^{8.3-5.2}=10^3.1, which is approximately 1258.92541179. So the San Francisco earthquake was approximately 1259 times more intense than the South America earthquake.
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