Problem 66 - College Algebra

Question

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula $$\log \frac{I_{1}}{I_{2}}=M_{1}-M_{2}$$ where \(M\) is the magnitude given by the Richter Scale. In August 2009 , an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011 , that same region experienced yet another, more devastating earthquake, this time with a magnitude of \(9.0 .^{[23]}\) How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Step-by-Step Solution

Answer

The 2011 earthquake was approximately 794 times more intense.

Step 1: Understand the Formula

The formula given to compare the intensity levels of two earthquakes is \( \log \frac{I_{1}}{I_{2}}=M_{1}-M_{2} \), where \( M \) is the magnitude on the Richter scale. \( I_1 \) and \( I_2 \) are the intensity levels of the two earthquakes.

Step 2: Substitute Known Values

Insert the given magnitudes into the formula, with \( M_1 = 9.0 \) for the March 2011 earthquake and \( M_2 = 6.1 \) for the August 2009 earthquake. This gives us the equation: \( \log \frac{I_1}{I_2} = 9.0 - 6.1 \).

Step 3: Calculate the Magnitude Difference

Calculate the difference \( M_1 - M_2 \):\[ 9.0 - 6.1 = 2.9 \].

Step 4: Set Up the Logarithmic Equation

From the equation, we have \( \log \frac{I_1}{I_2} = 2.9 \). This is the logarithmic form that relates the intensities of the two earthquakes.

Step 5: Convert to Exponential Form

To find \( \frac{I_1}{I_2} \), convert the logarithmic equation to its exponential form: \( \frac{I_1}{I_2} = 10^{2.9} \).

Step 6: Calculate the Intensity Ratio

Compute \( 10^{2.9} \) to find out how many times greater the intensity of the 2011 earthquake was compared to the 2009 earthquake: \( 10^{2.9} \approx 794.33 \).

Step 7: Round the Result

Round the calculated value \( 794.33 \) to the nearest whole number. The intensity of the 2011 earthquake was approximately 794 times greater than the 2009 earthquake.

Key Concepts

Earthquake Magnitude

The Richter Scale is a numerical scale used to express the magnitude, or size, of an earthquake. Created in 1935 by Charles F. Richter, this scale is logarithmic, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.
This scale is fundamentally important in seismology because it provides a standardized way to compare earthquakes in terms of their energy level or intensity.
When comparing the earthquakes of 2009 and 2011 in Honshu, the magnitude 6.1 and 9.0 respectively, represent the amount of energy released. Magnitude is a critical factor as it affects how much shaking and damage occurs in an earthquake. Understanding the specific magnitude helps scientists, engineers, and emergency responders better prepare for and respond to seismic events.

Logarithmic Functions

Logarithmic functions are essential in understanding phenomena that involve exponential growth or reduction, such as measuring earthquake intensity.
In a logarithmic equation, like \( \log \frac{I_1}{I_2} = M_1 - M_2 \), the logarithm (log) correlates the intensity ratio \( \frac{I_1}{I_2} \) to the difference in magnitudes, \( M_1 \) and \( M_2 \).
This logarithmic relationship means that a small change in magnitude results in a large change in intensity, which helps understand the significant impact of an increase in magnitude on the Richter Scale.
For instance, in our exercise, the magnitude difference is 2.9, which, when placed in the log equation, indicates a dramatic increase in energy release and corresponding intensity between the two earthquakes.

Exponential Form

Converting logarithmic equations to exponential form is vital in solving problems involving exponential relationships.
In our context, the equation \( \log \frac{I_1}{I_2} = 2.9 \) becomes \( \frac{I_1}{I_2} = 10^{2.9} \) when rewritten in exponential form. This transformation is key to quantifying the relationship between two differing magnitudes and their respective intensities.The exponential form clearly conveys the calculated ratio, showing that the intensity of the 2011 earthquake is approximately 794 times greater than that of the 2009 earthquake. This solidifies the understanding of how exponential growth, represented by powers of ten, reflects real-world phenomena like seismic intensity influenced by the Richter scale.