# Course:Math110/003/Teams/Ticino/The Richter Scale

## The Richter Scale

### Background

The Richter Scale is a logarithmic scale used to measure the strength and relative size of earthquakes. The earthquake in Chile last February measured an 8.8 on this scale, while the devastating Haitian earthquake was measured at 7.0.There is clearly a difference between these two numbers, but what does it really mean?

### Logarithmic functions

The Richter Scale measures how much the ground moves during an earthquake. It provides a single output number of seismic energy released during a quake. The formula for the Richter scale is as follows, where A is the amplitude of the readings recorded by the seismographs:

${\displaystyle M_{L}=\log _{10}A-\log _{10}A_{0}(\delta )}$[1]

We can use this to calculate the difference in terms of seismic energy released at the epicentre of the estimated earthquake versus the actual earthquake.

Magnitude estimated/Magnitude actual

We can then compare the estimate with the actual magnitude and often times we seen the estimate is off by a sufficient amount.

The logarithm of the amplitude is of the strongest wave and the magnitude is proportional to this.

From the following graph see that the Richter scale has an exponential growth.

The important thing to note here is that this formula uses a log of base 10. This means that for every increase of 1.0 in ${\displaystyle M_{L}}$, the amplitude of the ground's motion has increased by a factor of 10. Examining the previously-mentioned earthquakes in Chile and Haiti, the amplitude of the ground motion was 63 times greater in Chile than in Haiti. [4]

### Why Logarithmic?

Looking at this scale, one might wonder what the benefits of using a logarithmic scale. An increase of 1.0 does not intuitively suggest a tenfold increase in magnitude. However, if we look at the numbers involved, it makes a lot more sense.

Given that the amplitudes measured can vary so drastically, it makes sense to use a scale that deals in small numbers. The lowest rating on the scale that is typically felt by people is around 3.0. The "Big One" earthquake that is predicted for BC is expected to have a rating of approximately 9.0, or even higher. If we weren't using a logarithmic scale, we would be looking at something like the following:

3.0 on the scale: in the vicinity of ${\displaystyle 10^{3}}$, or 1,000.

9.0 on the scale: somewhere around ${\displaystyle 10^{9}}$, or 1,000,000,000.

It would be much more unwieldy talking about an earthquake that measured 532,054,942 on the amplitude scale. Being able to express this in logarithmic makes the information much more manageable and comprehensible to the average person. Logarithms take these large numbers and make them manageable to calculate as well as understand. In this sense, logarithms make understanding the severity of earthquakes in terms of numbers easier.