Homework 13 Group Work

The Logarithmic Scale and How it is applied to the Richter Scale

The logarithmic scale is used generally when there is a very wide range of values. It is important to note that the change of a specific value on according to a logarithmic scale does not depend on "the size of the change is proportion to the value of it self"(http://mathforum.org/library/drmath/view/55574.html) This can be explained easier if one looks at the difference between a linear and logarithmic scale. A linear scale is used "if adding 1 to a value is just a big as a change whether the original value was 1 or 1000"(http://mathforum.org/library/drmath/view/55574.html). In other words a linear scale is used when one can see on a graph the difference of an increase in 1(or any reasonably small number)

GRAPH OF LINEAR EQUATION

A logarithmic scale is used when the, "doubling of a value is just as big as a change whether it is from 1 to 2, or 1000 to 2000(http://mathforum.org/library/drmath/view/55574.html). Therefore the logarithmic scale was created and along the Y-Axis the number increase exponentially by ten each increasing value.

GRAPH OF LOGARITHMIC SCALE

In summation the logarithmic scale is used when there is a large range of values, and it does so by making each increasing value tenfold of the value preceding it. The logarithmic scale is used in a few instances including the Richter Scale.

The Richter magnitude scale

The Richter magnitude scale is used for the assigning of a numerical value to the seismic energy that is released by an earthquake. it is a base 10 logarithmic scale obtained by the calculation of the combined horizontal amplitude of the largest displacement from zero on a particular type of seismometer. (http://en.wikipedia.org/wiki/Richter_magnitude_scale)

The equation for the scale is

M_{\mathrm {L} }=\log _{10}A-\log _{10}A_{\mathrm {0} }(\delta )\

(A)\

derived from the the maximum excursion of the Wood-Anderson seismograph. The seismograph is used to measure the frequency of shaking the earth is going through while an earthquake is in progress.

(\delta )\

is the epicentra distance of the station which recoded the magnitude of the earthquake. The way that this distance is calculated is by measuring the time between P-waves and S-Waves at different stations that are located around the world. Once you have enough stations that have recorded these times one is able to triangulate the epicenter of the earthquake. (http://en.wikipedia.org/wiki/Epicentral_distance#Epicentral_distance)

Since this is a logarithmic function we know that an increase from 2 to 3 means that it is roughly 10 times stronger than its predecessor, in terms of earthquakes that means that the raw power that is released in that jump from 2 to 3 is about 31.6 times the energy released.

Conclusion

As stated above in the section regarding the Logarithmic Scale it does not depend on the absolute size of the change but according to the size of the change proportional to the initial value. So by applying this logic to that of earthquakes we have gathered information that shows how the power between a size 1 and a size 2 earthquake is about 31.6 times larger. This is jump in power can only be described in a logarithmic matter because as the size of the earthquake increases its power will always increase proportional to that of it.