The Richter scale is defined by the formula:
\[ M = \log_{10}\left(\frac{I}{I_0}\right) \]
where \( M \) is the magnitude of the earthquake, \( I \) is the intensity of the earthquake being measured, and \( I_0 \) is the intensity of a reference earthquake.
If the intensity of the earthquake is 10,000 times that of the reference earthquake, we can express this relationship as:
\[ I = 10,000 \times I_0 \]
Now, substituting this into the Richter scale formula gives:
\[ M = \log_{10}\left(\frac{10,000 \times I_0}{I_0}\right) \]
The \( I_0 \) cancels out:
\[ M = \log_{10}(10,000) \]
Since \( 10,000 \) is equal to \( 10^4 \):
\[ M = \log_{10}(10^4) = 4 \]
Therefore, the magnitude \( M \) of an earthquake with an intensity 10,000 times that of the reference earthquake is \( 4 \).
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