Problem 83 - Precalculus : Building Concepts and Connections

Question

Refer to the following. The magnitude of an earthquake is measured on the Richter scale using the formula $$R(I)=\log \left(\frac{I}{I_{0}}\right)$$ where I represents the actual intensity of the earthquake and \(I_{0}\) is a baseline intensity used for comparison. Richter Scale If the intensity of an earthquake is 10,000 times the baseline intensity \(I_{0},\) what is its magnitude on the Richter scale?

Step-by-Step Solution

Answer

The magnitude of the earthquake on the Richter Scale is 4.

Step 1: Understand and apply the formula

The problem provides the formula for calculating the Richter Scale: \(R(I)=\log \left(\frac{I}{I_{0}}\right)\). We know that the intensity \(I\) of a given earthquake is 10,000 times the baseline intensity \(I_{0}\). This information can be plugged into the formula, resulting in: \(R(I)=\log \left(\frac{10,000I_{0}}{I_{0}}\right)\).

Step 2: Simplify the equation

We simplify by dividing \(10,000I_{0}\) by \(I_{0}\), which results in \(10,000\). This simplifies our equation to \(R(I)=\log (10,000)\).

Step 3: Calculate the earthquake's magnitude on the Richter Scale

Using the definition of a logarithm, we know that \( \log(10,000)\) is asking for the exponent that 10 must be raised to in order to yield \(10,000\). Since \(10^4 = 10,000\), the exponent is \(4\), so the magnitude of the earthquake on the Richter Scale is \(4\).

Key Concepts

Logarithmic Functions

Understanding logarithmic functions is essential in various scientific fields, and one practical application of these functions is calculating the magnitude of earthquakes using the Richter scale formula. In simple terms, a logarithmic function is the inverse of an exponential function.

For example, the function \( y = \log_b(x) \) answers the question: to what power must the base \( b \) be raised to produce \( x \)? So, if you have an equation like \( 10^y = x \), then \( y = \log_{10}(x) \). Logarithms are invaluable in measuring quantities that cover a vast range of values, like earthquake intensities, because they allow us to convert multiplicative relationships into additive ones. This means we can work with manageable numbers to represent data that would otherwise be incomprehensibly large.

Calculating Earthquake Intensity

Measuring the intensity of an earthquake is crucial for assessing its potential damage, and this measurement is where the Richter scale comes into play. Developed by Charles F. Richter in 1935, the scale is logarithmic, meaning each whole number increase on the scale represents a tenfold increase in amplitude of the seismic waves and roughly 31.6 times more energy release.

Using the scale's formula \( R(I) = \log \left(\frac{I}{I_{0}}\right) \), we can calculate the magnitude of an earthquake. In this formula, \( I \) is the earthquake's intensity, and \( I_{0} \) is the baseline intensity. When an earthquake's intensity is a certain number of times greater than \( I_{0} \) (like 10,000 times in our example), its magnitude on the Richter scale can be found simply by taking the log of that number (in our case, the log of 10,000).

Applying Logarithms

The practical application of logarithms to calculate the magnitude of earthquakes on the Richter scale illustrates how logarithms help us deal with extraordinarily large or small numbers. When we applied the logarithmic function in our earthquake scenario, we were trying to figure out how much more intense this particular earthquake was compared to a base-level earthquake, a process which involved dividing the current intensity by the base intensity.

In our exercise, we simplified the equation by canceling out the common factors (the base intensities), which left us with the number 10,000. Recognizing that \( 10^4 = 10,000 \) makes the logarithm easy to compute, giving us a Richter scale magnitude of 4 for our example earthquake. By transforming the calculation into a question about exponents with logarithms, we unlock a straightforward and practical way of dealing with a complex natural phenomenon.