Question

3. In this section we have said that the Richter scale and other logarithmic scales are compressed. Explain in a way that a fellow student could understand what is meant by a compressed scale, and use a couple of examples to show how the compression works in practice.

Answer 1

Many measurements are on a special scale, the logarithmic scale.

A ruler is an example of a linear scale, where the distance between the 1 and 2 inch marks is the same as the distance between the 10 and 11 inch marks.

In a logarithmic scale, the unit steps increase in a multiplicative way. Consider the sequence:
1, 10, 100, 1000, ...

Each term is 10 times its immediate predecessor, which gives an exponential function . The function here would be f(t) = 10 ^t.

Decibel Scale

The human ear can hear sounds that are 100 trillion times louder than the faintest sounds. In the Decibel scale, the least audible sound, with an intensity of 10^-12watts/m² is assigned 0.
A sound 10 (= 10¹) times louder is assigned a decibel value of 10.
A sound 100 (= 10²) times louder is assigned a decibel value of 20.
A sound 1000 (= 10³) times louder is assigned a decibel value of 30.
and so on...

Problem:
If the sound of normal conversation is 60 decibels, and the sound in a subway is 90 decibels, how many times louder is a subway than a conversation?

Solution:
The difference in decibels is 90 - 60 = 30. Each increase of 10 decibels corresponds to 10 times the loudness. We have 10 * 10 * 10 = 1000, so the subway is 1000 times louder than normal conversation.

Richter Scale

Another logarithmic scale is the Richter Scale, used to measure magnitudes of earthquakes, since a unit change in the Richter Scale represents a tenfold increase in magnitude of the earthquake. If we use A for the response variable of amplitude of the earthquake, and r is the explanatory variable of the earthquake's magnitude, we have another exponential function, A= 10^r.

Problem:
How much stronger is an earthquake of magnitude 7.7 than one of 7.2?

Solution:
10^7.7 is approximately 50,118,723.4, and 10^7.2 is approximately 15,848,931.9.
So, 10^7.7 / 10^7.2 = 50,118,723.4 / 15,848,931.9, which is equal to 3.16. Then the earthquake measuring 7.7 on the Richter Scale is 3 times stronger than am earthquake measuring 7.2 on the Richter Scale.