MHB Richter Scale Earthquake Magnitude Question?

AI Thread Summary
The discussion revolves around calculating the Richter scale magnitude and energy release of earthquakes using the formula R = (2/3) log(E/E_0). The user seeks assistance in determining the magnitude of an earthquake that releases energy of E = 1000 E_0 and the energy released by a 5.0 magnitude earthquake, given E_0 = 10^{4.40}. They also need to find the energy ratio between an earthquake measuring 8.1 and an aftershock measuring 5.4 on the Richter scale. The calculations lead to various options for the correct answers, prompting the user to identify which one is accurate. The thread emphasizes the application of logarithmic identities in solving these problems.
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Hello. I am doing a review sheet for my Pre-Calculus final and one of the questions has me stumped. I'm going through our notes and we never did a problem like this in class. Any help would be greatly appreciated. Thank you.

Find the Richter scale magnitude of an earthquake that releases energy of E= 1000 E_0 . Then find the energy released by an earthquake that measures 5.0 on the Richter scale given that that E_0= 10^{4.40}. Finally find the ratio in energy released between an Earthquake that measures 8.1 on the Richter scale and an aftershock measuring 5.4 on the scale. Use the formula R = 2/3 log E/Eo

A) R = 2, E = 7.94 x 10^{11} joules and the ratio E1/E2 = 10200/1
B) R = 2, E = 7.94 X 10^{10} joules and the ratio E1/E2 = 11200/1
C) R = 2, E = 7.94 X 10^{11} joules and the ratio E1/E2 = 11200/1
D) R = 3, E = 7.94 X 10^{11} joules and the ratio E1/E2 11200/1
E) R = 2, E= 5.94 X 10^{11} joules and the ratio E1/E2 = 11200/1

Which answer would be the correct one?
 
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We are told to use the formula:

$$R=\frac{2}{3}\log\left(\frac{E}{E_0}\right)\tag{1}$$

For the first earthquake we are given, we are told $E=1000E_0=10^3E_0$. So, plugging this into (1), there results:

$$R=\frac{2}{3}\log\left(\frac{10^3E_0}{E_0}\right)=\frac{2}{3}\log\left(10^3\right)$$

Now, using the identities $\log_a\left(b^c\right)=c\log(b)$ and $\log_a(a)=1$, what do you find for $R$?
 
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