Richter Scale Earthquake Magnitude Question?

[kjeanch]

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?

 

[MarkFL]

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 $R$?