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. For any distance, D, we can solve for the illumination.Figure 1 Devastation of Maearthquake in Honshu, Japan.

This is an inverse square function – as distance increases at an exponential rate, illumination is divided by a number that is continually getting larger. To find a more precise relationship, we fit a curve to our points and the best equation to represent our data was “Illumination = A/(Distance)^2+B” where A=1.036e+005+/-771.7 and B=55.64+/-3.227. As the illumination approaches zero, the distance from the light source approaches infinity. Observing the graph, it is evident that as the distance from the light source grows, the brightness of the light decreases. The general equation relating light intensity and distance is is: Our specific equation we found when we fit a line to our data: The purpose of this lab was to determine a mathematical relationship between the intensity of a light and the distance from the light source, which we obtained by observing a graph. This provided a graphical representation comparing the brightness of light to distance away from the light source. We then collected our data, by measuring the light intensity at 1 cm increments from the light source up until the light sensor was 20cm from the source, at which point we switched to 10 cm increments all the way up to 110cm. In this lab, we used a light source to produce the light, and to measure the brightness of the light, we used the Vernier Light Sensor which determined the illumination (in lux). This happens because the closer you are to the source, the amount of light entering your eye increases.

When observing a light source, it appears to be brighter when you stand closer, and dimmer if you are positioned further away.

Light intensity equation exponent