Are the answers correct?

The Richter scale is used to measure the intensity of an earthquake. If E, measured in joules, is the energy released during an earthquake, then the magnitude is given by,

M = 2/3 log(E/E_0)
Where E_0 = 10^4.4 joules

1. If 8 x 10^14 joules of energy are released during an earthquake, what is the magnitude of the earthquake?

answer: M = 2/3 log_10((8×10^14)/(1×10^4.4)) =~7 - Magnitude on the Richter scale

2. How much energy will be released in an earthquake with a magnitude of 5.9?

Answer: 5.9 = 2/3 log_10(P/(1×10^4.4))
P = 1.77828×10^13
- Joules - the amount of energy released in a 5.9 Magnitude earthquake.

first one around 2/3 of around 11 which is around 7, ok

9 = log (about 10^9 )check

To check if the answers provided are correct, let's go through the calculations step-by-step:

1. To find the magnitude of an earthquake when given the energy released (E), we use the formula:
M = (2/3) * log10(E/E_0), where E_0 = 10^4.4 joules.

So, for the first question, where E = 8 x 10^14 joules, we substitute the values in the formula:
M = (2/3) * log10((8 × 10^14)/(1 × 10^4.4))
M = (2/3) * log10(8 × 10^14/10^4.4)
M = (2/3) * log10(8 × 10^14/2.511886432) (simplifying 10^4.4 ≈ 2.511886432)
M = (2/3) * log10(3.183098861 × 10^14)
M ≈ 2/3 * 14.5
M ≈ 9.67

Hence, the magnitude of the earthquake is approximately 9.67 on the Richter scale, not 7 as stated in the answer.

2. To find the energy released (E) in an earthquake when given the magnitude (M), we rearrange the formula:
M = (2/3) * log10(E/E_0) in terms of E.

So, for the second question, where M = 5.9, we rearrange the formula:
5.9 = (2/3) * log10(P/(1 × 10^4.4))
where P represents the energy released.

To solve for P, we can isolate it:
5.9 * (3/2) = log10(P/(1 × 10^4.4))
8.85 = log10(P/(1 × 10^4.4))

Now, we need to use logarithmic properties to rewrite the equation without the logarithm:
10^8.85 = P/(1 × 10^4.4)
10^(8.85 + 4.4) = P
10^13.25 = P

Hence, the amount of energy released in an earthquake with a magnitude of 5.9 is approximately 10^13.25 joules, not 1.77828×10^13 joules as stated in the answer.

Therefore, the answers provided are not correct.