Problem 367 - Precalculus

Question

The magnitude \(M\) of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where \(E\) is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4\(\cdot 10^{13}\) joules of energy?

Step-by-Step Solution

Answer

The magnitude of the earthquake is approximately 5.83.

Step 1: Identify Given Values

First, we need to identify the given values from the problem statement. We are given that the energy released by the earthquake, \(E\), is \(1.4 \cdot 10^{13}\) joules, and \(E_{0}\) is \(10^{4.4}\) joules.

Step 2: Substitute Values in the Formula

Plug the given values for \(E\) and \(E_{0}\) into the magnitude formula: \[ M = \frac{2}{3} \log \left(\frac{1.4 \cdot 10^{13}}{10^{4.4}}\right) \]

Step 3: Simplify the Expression

Calculate the exponent part inside the logarithm: \[ \frac{1.4 \cdot 10^{13}}{10^{4.4}} = 1.4 \cdot 10^{13 - 4.4} = 1.4 \cdot 10^{8.6} \]

Step 4: Compute the Logarithm

Calculate the logarithm of the simplified expression: \[ \log(1.4 \cdot 10^{8.6}) = \log(1.4) + \log(10^{8.6}) = \log(1.4) + 8.6 \] Given \(\log(1.4) \approx 0.146\), we find: \[ \log(1.4 \cdot 10^{8.6}) = 0.146 + 8.6 = 8.746 \]

Step 5: Calculate the Magnitude

Now substitute back into the equation to find the magnitude \(M\): \[ M = \frac{2}{3} \cdot 8.746 \approx \frac{17.492}{3} \approx 5.831 \]

Step 6: Round to Nearest Hundredth

Round the calculated magnitude to the nearest hundredth: \[ M \approx 5.83 \]

Key Concepts

Logarithmic Scale

Earthquakes are measured using a logarithmic scale, which is a way to compare numbers using powers of ten. This means that each whole number increase in magnitude represents a tenfold increase in the energy released. For example, an earthquake with a magnitude of 6.0 releases ten times more energy than one of magnitude 5.0.

This scale is particularly useful in geology because of the vast range of earthquake energies. Without it, comparing the energy released by different earthquakes would be much more cumbersome. The logarithmic scale simplifies this by converting these large differences into manageable numbers.

It helps us understand the relative size of earthquakes in a way that is more intuitive than working directly with the absolute energy values.

Energy Release

The energy released during an earthquake is a key factor in determining its magnitude. This energy is measured in joules, a unit of energy in the International System of Units (SI). In the context of earthquakes, higher energy release means a more powerful and usually more destructive earthquake.

In our exercise, the energy release is given as 1.4 × 10¹³ joules, which is quite significant. This number reflects how much energy is being transferred from the earth's movements.

• High energy release: leads to higher magnitudes and potential damage.
• Low energy release: lesser magnitude, often barely felt.

Understanding energy release is crucial for assessing the impact of an earthquake.

Natural Logarithm

A natural logarithm is a type of logarithm that uses the mathematical constant e (approximately 2.718) as its base. While our exercise uses the common logarithm, which is base 10, understanding logarithms in general, including the natural logarithm, is important.

Logarithms transform multiplicative relationships into additive ones, making it easier to manage large numbers. In earthquake calculations, it helps convert the massive differences in energy releases into smaller, more practical numbers.

The use of natural logarithms in science is similar to the way common logarithms are used in earthquake magnitude calculations, in that they both serve to simplify complex multiplicative scenarios. Recognizing this relationship can help demystify how earthquakes are measured.

Magnitude Formula

The magnitude formula used in the exercise is a mathematical representation of how we calculate an earthquake's magnitude based on energy release.The formula is:\[ M = \frac{2}{3} \log \left(\frac{E}{E_0}\right) \]- **M**: The magnitude of the earthquake.- **E**: The energy released, measured in joules.- **E₀**: A constant reference energy level, which is defined as 10⁴⋅⁴ joules in this instance.By substituting the energy figures into the formula, you calculate the magnitude. The formula's use of logarithms allows it to effectively manage the vast range of energy values into numbers that make practical sense.

This is why the magnitude of earthquakes can be easily communicated and compared, helping communities and scientists recognize the potential severity of seismic events.