Problem 43 - Intermediate Algebra

Question

An earthquake in Los Angeles in 1971 had an intensity of approximately five million times the reference intensity. What was the Richter number associated with that earthquake?

Step-by-Step Solution

Answer

The Richter magnitude of the 1971 Los Angeles earthquake was approximately 6.7.

Step 1: Understand the Richter Scale

The Richter scale is logarithmic, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude and approximately 31.6 times more energy released. It uses the formula: \[ M = \log_{10}\left( \frac{I}{I_0} \right) \]where \( M \) is the magnitude on the Richter scale, \( I \) is the intensity of the earthquake, and \( I_0 \) is the reference intensity.

Step 2: Apply the Formula

Given that the intensity \( I \) is five million times the reference intensity \( I_0 \), we can express the intensity as \( I = 5,000,000 \times I_0 \). Substitute into the formula:\[ M = \log_{10}\left( \frac{5,000,000 \times I_0}{I_0} \right) = \log_{10}(5,000,000) \]

Step 3: Calculate the Logarithm

Calculate the base-10 logarithm of 5,000,000:\[ \log_{10}(5,000,000) = \log_{10}(5 \times 10^6) \]Apply the logarithm of a product: \[ \log_{10}(5) + \log_{10}(10^6) \]This becomes:\[ \log_{10}(5) + 6 \approx 0.7 + 6 \]

Step 4: Determine the Richter Magnitude

Add the calculated values to find:\[ M \approx 0.7 + 6 = 6.7 \]Thus, the magnitude of the earthquake on the Richter scale is approximately 6.7.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, particularly when dealing with exponential growth or decay. A logarithm answers the question: "To what power must a certain base be raised to produce a given number?" For example, in base-10 logarithms, also known as common logarithms, \[ \log_{10}(100) = 2 \]because 10 raised to the power of 2 equals 100.
Logarithms turn multiplication into addition, which simplifies many calculations and problems involving exponential relationships. They are especially useful in scientific fields like seismology, where they help express large scales of variation in intensity or magnitude simply.
In the context of earthquakes, the Richter scale employs logarithmic calculations to make it easier to compare earthquake intensities that vary widely. Instead of discussing millions of times in amplitude, we can talk about it in a more manageable scale of numbers, such as 6.7. This scale allows us to recognize that an increase of 1 in magnitude represents about 31.6 times more energy released, making significant differences easily distinguishable.

Earthquake Intensity

Understanding earthquake intensity is crucial for comprehending how deeply and broadly an earthquake can affect regions. Intensity measures how much the ground shakes at a particular location due to the earthquake, which can affect human structures and terrain.
Unlike magnitude, which is a measure of the energy released, intensity is more about the observed effects and vibration impact. The reference intensity is the baseline measurement from which other intensities are compared. In the provided example, the reference intensity is what the intensity of the 1971 Los Angeles earthquake was compared to, indicating that it was 5,000,000 times stronger than this reference level.
The statement that the intensity is 5,000,000 times the reference is crucial because it informs us of the magnitude using logarithms. Understanding intensity helps governments and planners make informed decisions on building infrastructure meant to withstand earthquakes and manage emergency responses effectively.

Magnitude Calculation

Calculating the magnitude of an earthquake helps determine its potential destructive power. The magnitude is computed using a logarithmic formula based on the intensity of the earthquake compared to a reference intensity. This is expressed in the formula:\[ M = \log_{10}\left( \frac{I}{I_0} \right) \]
In this formula:

• \( M \) represents the magnitude on the Richter scale.
• \( I \) is the actual intensity of the earthquake.
• \( I_0 \) is the reference intensity.

To calculate the magnitude of the 1971 Los Angeles earthquake, we first note that its intensity was 5,000,000 times the reference intensity. Thus:\[ M = \log_{10}\left( \frac{5,000,000 \times I_0}{I_0} \right) = \log_{10}(5,000,000)\]
Breaking down the calculation, \[5,000,000 = 5 \times 10^6\], thus:\[ \log_{10}(5,000,000) = \log_{10}(5) + \log_{10}(10^6) = \log_{10}(5) + 6 \]With \(\log_{10}(5)\) approximately 0.7, the magnitude \(M\) is approximately:\[ 6.7 \]
This illustrates the enormous potential impact of the earthquake, characterized by this single scalar number known as the Richter magnitude.