The base b b logarithm of a number is the exponent by which we must raise b b to get that number. To represent y y as a function of x, x, we use a logarithmic function of the form y = log b ( x ). As is the case with all inverse functions, we simply interchange x x and y y and solve for y y to find the inverse function. The exponential function y = b x y = b x is one-to-one, so its inverse, x = b y x = b y is also a function. Observe that the graph in Figure 2 passes the horizontal line test. To find an algebraic solution, we must introduce a new function. We can examine a graph, as in Figure 2, to better estimate the solution.Įstimating from a graph, however, is imprecise. We know that 10 2 = 100 10 2 = 100 and 10 3 = 1000, 10 3 = 1000, so it is clear that x x must be some value between 2 and 3, since y = 10 x y = 10 x is increasing. None of the algebraic tools discussed so far is sufficient to solve 10 x = 500. We have not yet learned a method for solving exponential equations. The equation that represents this problem is 10 x = 500, 10 x = 500, where x x represents the difference in magnitudes on the Richter Scale. We want to calculate the difference in magnitude. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. Converting from Logarithmic to Exponential Form It is 10 8 − 4 = 10 4 = 10,000 10 8 − 4 = 10 4 = 10,000 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. The Richter Scale is a base-ten logarithmic scale.
The Haitian earthquake registered a 7.0 on the Richter Scale 6 whereas the Japanese earthquake registered a 9.0. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, 5 like those shown in Figure 1. In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes 4. Figure 1 Devastation of Maearthquake in Honshu, Japan.
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