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Chapter 4: Problem 35

ln 1986, there was an earthquake near Cleveland, Ohio. It had an intensity of \(10^{5} \cdot I_{0} .\) What was its magnitude on the Richter scale?

Short Answer

Expert verified
The magnitude on the Richter scale is 5.

Step by step solution

01

- Understand the Richter Scale Formula

The magnitude on the Richter scale is given by the formula \[ M = \text{log}_{10} \frac{I}{I_{0}} \] where \( M \) is the magnitude, \( I \) is the intensity of the earthquake, and \( I_{0} \) is a reference intensity.
02

- Substitute the Given Values

Substitute the given intensity into the formula. We are given that the intensity of the earthquake is \( 10^{5} \times I_0 \). Hence, \( I = 10^{5} \times I_0 \).
03

- Simplify the Equation

Simplify the equation by substituting \( I = 10^{5} \times I_0 \) into the formula: \[ M = \text{log}_{10} \frac{10^{5} \times I_{0}}{I_{0}} \]
04

- Calculate the Magnitude

Since \( I_{0} \) in the numerator and the denominator cancels each other out, the equation simplifies to: \[ M = \text{log}_{10} 10^{5} \] Using the property of logarithms that \( \text{log}_{10} 10^{x} = x \), we get: \[ M = 5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithms
Logarithms are a mathematical concept that helps us deal with very large or very small numbers more easily. A logarithm answers the question:

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