In order to solve problems involving logarithmic models, it is necessary to
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
) 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
. In the formula, x
stands for the intensity
of the earthquake
represents the seismographic reading in millimeters. The xo
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 xo:
Therefore, the magnitude of the earthquake
that hit San Francisco in 1906 was 6.