Problem 35 - Algebra and Trigonometry

Question

\(29-43\) . These exercises deal with logarithmic scales. Earthquake Magnitudes If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

Step-by-Step Solution

Answer

Its magnitude is about 1.301 units larger on the Richter scale.

Step 1: Understand the Richter Scale

The Richter scale is a logarithmic scale used to measure the magnitude of earthquakes. Since it is a base-10 logarithmic scale, an increase of 1 unit on the Richter scale corresponds to a tenfold increase in measured amplitude and roughly 31.6 times more energy release.

Step 2: Express the Intensity Relation

We are given that one earthquake is 20 times as intense as another. Therefore, if the intensity of the first earthquake is \(I_1\) and the second is \(I_2\), then \(I_1 = 20 \times I_2\).

Step 3: Use the Richter Scale Formula

The magnitude \(M\) on the Richter scale is given by \(M = \log_{10}(I/I_0)\), where \(I\) is the intensity and \(I_0\) is some reference intensity. Thus, the difference in magnitude \(\Delta M\) is \(\Delta M = \log_{10}(I_1/I_0) - \log_{10}(I_2/I_0)\).

Step 4: Simplify the Magnitude Difference

Using the properties of logarithms, specifically \(\log(a) - \log(b) = \log(a/b)\), we have:\[\Delta M = \log_{10}\left(\frac{I_1}{I_2}\right) = \log_{10}(20)\].

Step 5: Calculate the Logarithm

Now, calculate \(\log_{10}(20)\). You can use a calculator to find that \(\log_{10}(20) \approx 1.301\).

Step 6: State the Result

The difference in magnitude \(\Delta M\) is approximately 1.301, meaning the earthquake that is 20 times as intense has a magnitude that is about 1.301 units higher on the Richter scale.

Key Concepts

Logarithmic Scale

When dealing with measurements that cover a vast range of values, a logarithmic scale becomes particularly useful. Unlike linear scales, where each unit represents an equal increment, logarithmic scales measure quantities in multiplicative increments. This means that each step on the scale corresponds to a multiplication of the value.
For example, on a base-10 logarithmic scale, each whole number increase on the scale represents a tenfold increase in actual value. The Richter scale, used for measuring earthquakes, is a perfect example of this principle in action.

• Logarithmic scales are instrumental in sciences, such as geology and astronomy, where values can span multiple orders of magnitude.
• They help us represent large numbers in a more manageable way, aiding in visualization and comparison.

Understanding how to work with logarithms is crucial when interpreting these scales, allowing us to grasp the underlying magnitude changes.

Earthquake Intensity

Earthquake intensity refers to the strength or force of an earthquake at a particular location. This can vary depending on the distance from the earthquake's epicenter. The key point is that intensity quantifies the energy released by the earthquake, typically reflecting its potential for damage.
With earthquakes, small increases in intensity can lead to disproportionately large effects in terms of energy release. For instance, an earthquake that is 20 times more intense than another will not only shake the ground harder but will also release a vastly greater amount of energy.

• Intensity is often measured in terms of acceleration, velocity, or displacement of the ground.
• It gives an idea of how powerful the earthquake feels at different places.

Numerous scales exist to measure intensity, but they can get confused with the magnitude. Remember, intensity reflects local effects, while magnitude reflects the total energy released.

Richter Scale Magnitude

The Richter scale is specifically designed to measure the magnitude of earthquakes. This scale assigns a single number to quantify the energy released by an earthquake, unlike intensity, which can differ by location.
The peculiar thing about the Richter scale is that it's a logarithmic scale. A 1-unit increase on this scale signals a tenfold increase in amplitude and roughly 31 times more energy release. Therefore, even small differences in magnitude can indicate significant differences in the earthquake's actual power and potential for destruction.

• An earthquake with a Richter scale reading of 6.0 releases about 31 times more energy than one with 5.0.
• Magnitude does not depend on the location or structure of an observer; it is a fundamental measure of the energy released.

The predictability and universality of the Richter scale make it invaluable in both scientific research and public information.

Properties of Logarithms

Logarithms have several key properties that make them incredibly useful, especially in dealing with exponential processes like earthquakes. One fundamental property is their ability to transform multiplication into addition. For example, o \ a base-10 logarithm of a quotient, such as \( \log_{10}( \frac{a}{b} ) \), can be expressed as \( \log_{10}(a) - \log_{10}(b) \).
This property simplifies the process of comparing two values by reducing them to their logarithmic difference.

• Logarithms allow for computations involving very large or small numbers to be executed more succinctly.
• They are defined as the inverse operation to exponentiation, meaning \( y = \log_{b}(x) \) is equivalent to \( b^y = x \).
• This makes them invaluable in solving equations where the variable is an exponent.

Making sense of how to break down and utilize these properties is vital in calculations across numerous scientific fields, including seismology when using the Richter scale.