A lot of important  physical quantities , such as energy released in an earthquake , have very large ranges
 and thus logarithmic scales( the Richter Scale, in the case of an earthquake)are helpful in providing a
 way of making the numerical values of these measurements manageable.

 Scientists have proven that an increase of 0,2 on the Richter Scale roughly doubles the amount of energy released by an earthquake.
An increase of 1.0 gives a multiplying factor of 31.7 ( very close to 2`).


An earthquake of magnitude 3.0 on the Richter Scale releases an energy equivalent to 480kg of TNT.
a) How much energy will be released by an earthquake of the following magnitudes 

3.2
4.0
7.5


Calculate using logarithms

To solve this problem, we'll use the information given about the Richter scale and logarithms. We know that:

- An increase of 0.2 on the Richter scale doubles the energy released by an earthquake.
- An increase of 1.0 on the Richter scale multiplies the energy released by a factor of 31.7.

### Step 1: Find the relationship between magnitude and energy

Let \( E(M) \) represent the energy released by an earthquake of magnitude \( M \) on the Richter scale. We know:

- \( E(3.0) = 480 \) kg of TNT.
- \( E(M) \) is proportional to \( 31.7^{M-3.0} \).

So, we can express the energy \( E(M) \) as:
\[
E(M) = 480 \times 31.7^{(M-3.0)}
\]

### Step 2: Calculate the energy released for each magnitude

#### a) Magnitude 3.2

Using the formula:
\[
E(3.2) = 480 \times 31.7^{(3.2-3.0)}
\]

\[
E(3.2) = 480 \times 31.7^{0.2}
\]

We know that an increase of 0.2 on the Richter scale doubles the energy:
\[
E(3.2) = 480 \times 2 = 960 \text{ kg of TNT}
\]

#### b) Magnitude 4.0

Using the formula:
\[
E(4.0) = 480 \times 31.7^{(4.0-3.0)}
\]

\[
E(4.0) = 480 \times 31.7^{1.0} = 480 \times 31.7
\]

\[
E(4.0) \approx 480 \times 31.7 = 15216 \text{ kg of TNT}
\]

#### c) Magnitude 7.5

Using the formula:
\[
E(7.5) = 480 \times 31.7^{(7.5-3.0)}
\]

\[
E(7.5) = 480 \times 31.7^{4.5}
\]

We can calculate this as:
\[
E(7.5) = 480 \times 31.7^{4.5} \approx 480 \times 1613541.2 \approx 774,429,376 \text{ kg of TNT}
\]

### Summary of Results:
- Magnitude 3.2 releases approximately **960 kg** of TNT.
- Magnitude 4.0 releases approximately **15,216 kg** of TNT.
- Magnitude 7.5 releases approximately **774,429,376 kg** of TNT.

Would you like a detailed breakdown of the logarithmic steps? 

Here are 5 related questions to explore:
1. How is the logarithmic scale defined mathematically?
2. How does the Richter scale compare to other earthquake magnitude scales?
3. What are the implications of using a logarithmic scale in measuring physical quantities?
4. How does energy release in an earthquake translate to ground movement and damage?
5. Can we use logarithms to model other natural phenomena?

**Tip:** When working with logarithms, always ensure the base and the relationship between quantities are clearly understood to avoid mistakes in calculation.

Learn how to calculate the energy released by earthquakes of different magnitudes using logarithmic scales. This problem demonstrates the use of logarithms in quantifying exponential growth in energy release. Suitable for high school students (Grades 10-12).

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