Logarithmic models are useful in several physical applications including the following: magnitude of earthquakes, intensity of sound, and acidity of a solution. A logarithmic model generally has a period of rapid increase followed by a period of slow growth, but the model continues infinitely without bound.

One example of a logarithmic modeling problem involves finding the magnitude of an earthquake. The Richter scale is a common method used to measure the intensity of an earthquake. The scale converts seismographic readings into numbers that offer an easy reference for measuring the magnitude (M) of an earthquake. All earthquakes are compared to a zero-level earthquake (x_{o}) whose seismographic reading measures 0.001 millimeter at a distance of 100 kilometers from the epicenter.

The formula used to find the measure of the magnitude of an earthquake is: . In the formula, x stands for the intensity of the earthquake and x represents the seismographic reading in millimeters. The x_{o }represents a zero-level earthquake the same distance from the epicenter.

Suppose that you wanted to find the magnitude of the San Francisco Earthquake of 1906 given the data that a seismographic reading of 7,943 millimeters was registered 100 kilometers from the center.

First, we will need to use the logarithmic models formula for finding magnitude of an earthquake:

In the formula, we substitute 7943 mm for the value of x, and 0.001 for the value of x_{o}:

Now, we type into a calculator to evaluate the function and find M(x):

Therefore, the magnitude of the earthquake that hit San Francisco in 1906 was 6.9.

Let's Practice

Question #1

Biologists use the logarithmic model to estimate the number of species (n) that live in a region of area (A). In the model, k represents a constant. If 2800 species live in a rain forest of 500 square kilometers, then how many species will be left when half of this rain forest is destroyed by logging?

Question #2

The intensity of a sound wave is interpreted by our ears as its loudness. The weakest sound wave that a human ear can hear has a value of 1 x 10^{-12} watts/m^{2} and is called the threshold of human hearing, I_{o}.

To compare relative sound intensities, we use a scale called decibels, dB, which is calculated with the formula . In this formula, the threshold of human hearing would have a decibel reading of 0 dB.

What is the intensity in watt/m^{2} of a sound wave that has a sound level reading of 125 dB, the loudness of an average fire alarm?

Question #3

Desalination is the process of producing fresh water from salt water. How much fresh water can be produced after 10 hours from a desalination process using the formula , given that after 1 hour, 18.27 cubic yards of fresh water can be produced and that the rate of production for the desalination process is 31.03?

Before beginning our solution, examine the function to see that:

y represents the amount of fresh water to be produced as time passes,

t represents time in hours,

b is a number that controls the rate of production of fresh water, and

a represents the amount of fresh water produced after 1 unit of time has passed.

All functions of this type increase without bound to the right, pass through the point (1,a), and grow rapidly in the beginning and then grow very slowly. The rate of growth for the function is controlled by b. Take a moment and familiarize yourself with the properties of this function in the illustrations below.

y = 1 + lnx

y = 3 + 2 lnx

y = -2 + 2 lnx

Try These

Question #1

If 3600 species live in a rain forest of 700 square kilometers, then how many species will be left when three-fourths of the forest is destroyed by logging?

Mexico City had an earthquake in 1985 that had a seismographic reading of 125,892 millimeters 100 kilometers from the center. Find the magnitude of the earthquake.

A. 0.005

B. 2.10

C. 8.1

D. 7.10

As you can see, this type of problem requires that you write a logarithmic modeling formula based on given information. You must then correctly substitute given values for variables and solve the equation you obtain. At the conclusion of the problem, you should always check for the reasonableness of your solution.