Problem 25 - College Algebra

Question

Recall the formula for calculating the magnitude of an earthquake, \(M=\frac{2}{3} \log \left(\frac{S}{S_{0}}\right) .\) Show each step for solving this equation algebraically for the seismic moment \(S\)

Step-by-Step Solution

Answer

Solve for the seismic moment using the formula: \(S = S_{0} \cdot 10^{\frac{3}{2}M}\).

Step 1: Understand the Given Equation

The formula for calculating the magnitude \(M\) of an earthquake is given by \(M = \frac{2}{3} \log \left(\frac{S}{S_{0}}\right)\), where \(S\) is the seismic moment we want to solve for, and \(S_0\) is the reference seismic moment.

Step 2: Isolate the Logarithmic Term

First, begin by eliminating the fraction. Multiply both sides of the equation by \(\frac{3}{2}\) to start isolating the logarithmic term: \[ \frac{3}{2}M = \log \left(\frac{S}{S_{0}}\right) \]

Step 3: Exponentiate to Solve for Seismic Moment Ratio

To eliminate the logarithm, exponentiate both sides of the equation. Recall that if \(\log(a) = b\), then \(a = 10^b\). Thus, \[ 10^{\frac{3}{2}M} = \frac{S}{S_{0}} \]

Step 4: Solve for Seismic Moment \(S\)

Multiply both sides of the equation by \(S_0\) to solve for \(S\): \[ S = S_{0} \cdot 10^{\frac{3}{2}M} \]

Step 5: Interpret the Final Solution

The expression \(S = S_{0} \cdot 10^{\frac{3}{2}M}\) provides the relationship between the seismic moment \(S\), magnitude \(M\), and reference seismic moment \(S_0\). This formula allows us to find \(S\) if both \(S_0\) and \(M\) are known.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in algebra, serving as the inverse operations of exponentiation. A logarithm answers the question: "To what power must a specific base be raised, in order to produce a given number?" In the context of logarithmic expressions, such as \( \log(b) = a \), the base is usually either 10 (common logarithm) or \( e \) (natural logarithm). Logarithms simplify multiplication and division into addition and subtraction.

• This property arises because of the laws of exponents, where multiplying powers of the same base results in adding their exponents.
• Using logarithms can significantly ease complex calculations, making them useful in fields like seismology, as evidenced by earthquake magnitude equations.

The logarithm of a number with respect to a certain base is only defined for positive numbers. This is why understanding the behavior of logarithmic functions and knowing their rules, such as the power rule, product rule, and change of base formula, is crucial.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. This operation is denoted as \( b^n \), where \( b \) is the base and \( n \) is the exponent. The expression represents the base being multiplied by itself \( n \) times. Exponentiation is the inverse operation of logarithms. It allows us to convert logarithmic equations back into their exponential form, facilitating the solving process.

• This is particularly useful when solving equations like those involving earthquake magnitudes, where you isolate a variable within a logarithmic function.
• The equation \( 10^{\frac{3}{2}M} = \frac{S}{S_{0}} \) arises from engaging exponentiation to solve for the ratio in the logarithmic earthquake formula.

Exponentiation also comes with its set of rules such as the power rule, product rule, and quotient rule, each allowing for manipulation and simplification of expressions involving powers and roots.

Earthquake Magnitude

The magnitude of an earthquake is a measure of the energy released at the source of the earthquake. It is commonly expressed using a logarithmic scale because of the wide range of possible energy values. Earthquakes can release energy that spans several orders of magnitude, making a linear scale impractical.

• Using the equation \( M = \frac{2}{3} \log \left(\frac{S}{S_{0}}\right) \), seismologists can quantify the size of an earthquake relative to a reference or baseline.
• This equation means that each whole number increase on the logarithmic scale corresponds to a tenfold increase in measured amplitude and about 31.6 times more energy release.

Understanding earthquake magnitude is crucial for assessing potential damage and for planning disaster prevention and response strategies.

Seismic Moment

Seismic moment, denoted as \( S \), is a measure of an earthquake's total energy. It represents the actual physical process occurring along a fault during an earthquake. The seismic moment is determined by several factors: the amount of slip on the fault, the area of the fault that slipped, and the strength of the rocks involved.

• The seismic moment is directly related to the magnitude of an earthquake; a larger seismic moment indicates a larger release of energy and typically a higher magnitude.
• The equation \( S = S_{0} \cdot 10^{\frac{3}{2}M} \) provides a way to calculate the seismic moment from a known magnitude and reference seismic moment \( S_0 \).

Grasping the concept of seismic moment is essential for understanding the mechanics of earthquakes and their potential impacts.