Answered

The Richter scale measures the magnitude, $M$, of an earthquake as a function of its intensity, $I$, and the intensity of a reference earthquake, $I_0$.



$M=\log \left(\frac{I}{I_0}\right)$



Which equation calculates the magnitude of an earthquake with an intensity 10,000 times that of the reference earthquake?


A$M=\log \left(\frac{I}{10,000}\right)$
B$M=\log \left(\frac{1}{10,000}\right)$
C$M=\log \left(\frac{10,000}{I_0}\right)$
D$M=\log (10,000)$

Verified Answer :

  • The formula for earthquake magnitude is \(M = \log \left(\frac{I}{I_0}\right)\).
  • Given that the intensity \(I = 10,000 \cdot I_0\), we substitute this into the formula.
  • Simplifying the fraction \(\frac{10,000 \cdot I_0}{I_0}\) results in \(10,000\).
  • The final expression for the magnitude is \(\boxed{\log (10,000)}\).

Explanation

  1. Understanding the Problem and Data Analysis

The Richter scale magnitude \(M\) is defined by the formula \[M = \log \left(\frac{I}{I_0}\right)\] where \(I\) represents the intensity of the earthquake and \(I_0\) is the intensity of a reference earthquake. We are given the information that the earthquake's intensity \(I\) is 10,000 times that of the reference earthquake, which can be expressed as:
\[I = 10,000 \cdot I_0\]
Our objective is to find the expression for the magnitude \(M\) by substituting this relationship into the given formula.

  1. Calculating the Magnitude of the Earthquake

To find the magnitude \(M\) for an earthquake with an intensity 10,000 times greater than the reference intensity, we substitute the known relationship \(I = 10,000 \cdot I_0\) into the formula for \(M\):
\[M = \log \left(\frac{10,000 \cdot I_0}{I_0}\right)\]
Now, we simplify the expression inside the logarithm by canceling out the common term \(I_0\) from the numerator and the denominator, assuming \(I_0 eq 0\):
\[M = \log (10,000)\]
This result represents the magnitude of the earthquake. Comparing this with the given choices, we see that it matches the fourth option.

Examples

The Richter scale helps seismologists communicate the severity of earthquakes to the public in a standardized way. For instance, if a major earthquake registers a magnitude of 7.0 and a minor one registers 3.0, the difference in magnitude is \(7-3=4\). Because the scale is logarithmic, this means the major earthquake is \(10^4 = 10,000\) times more intense than the minor one. Understanding how these logarithmic relationships work allows scientists to analyze and report the relative energy release of seismic events efficiently.

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