Charles Richter defined the magnitude of an earthquake to be ` M = log_(10) (I/S)`, where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicentre of the earthquake) and S is the intensity of a ''standed earthquake'' (whose amplitude is 1 micron `=10^(-1)` cm).
Each number increase on the Richter scale indicates an intensity ten times stronger. For example. an earthquake of magnitude 5. An earthquake of magnitude 7 is 100 times stronger then an earthquake of magnitude 5. An earthquake of magnitude 8 is 1000 times stronger than an earthquake of magnitude 5.
The earthquake in city A registered `8.3` on the Richter scale. In the same year, another earthquake was recorded in city B that was four times stronger. What was the magnitude of the earthquake in city B ?

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that the magnitude of an earthquake is given by the formula:
\[ M = \log_{10} \left( \frac{I}{S} \right) \]
where \( I \) is the intensity of the earthquake and \( S \) is the intensity of a standard earthquake.

### Step 2: Set up the equation for city A
The earthquake in city A registered a magnitude of \( 8.3 \). We can denote the intensity of the earthquake in city A as \( I_A \). Thus, we can write:
\[ M_A = \log_{10} \left( \frac{I_A}{S} \right) \]
Given that \( M_A = 8.3 \), we have:
\[ 8.3 = \log_{10} \left( \frac{I_A}{S} \right) \]
This is our **Equation 1**.

### Step 3: Express the intensity of city B
We are told that the earthquake in city B is four times stronger than that in city A. Therefore, we can express the intensity of the earthquake in city B as:
\[ I_B = 4 I_A \]

### Step 4: Set up the equation for city B
Now we need to find the magnitude of the earthquake in city B, denoted as \( M_B \):
\[ M_B = \log_{10} \left( \frac{I_B}{S} \right) \]
Substituting \( I_B = 4 I_A \) into the equation, we get:
\[ M_B = \log_{10} \left( \frac{4 I_A}{S} \right) \]

### Step 5: Use properties of logarithms
Using the property of logarithms that states \( \log_{10}(a \cdot b) = \log_{10}(a) + \log_{10}(b) \), we can rewrite the equation for \( M_B \):
\[ M_B = \log_{10}(4) + \log_{10} \left( \frac{I_A}{S} \right) \]

### Step 6: Substitute from Equation 1
From **Equation 1**, we know that:
\[ \log_{10} \left( \frac{I_A}{S} \right) = 8.3 \]
Thus, we can substitute this into the equation for \( M_B \):
\[ M_B = \log_{10}(4) + 8.3 \]

### Step 7: Calculate \( \log_{10}(4) \)
The value of \( \log_{10}(4) \) is approximately \( 0.602 \). Therefore:
\[ M_B = 0.602 + 8.3 \]

### Step 8: Final calculation
Now, we can perform the final addition:
\[ M_B = 8.902 \]

### Conclusion
The magnitude of the earthquake in city B is approximately:
\[ \boxed{8.902} \]

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