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 (exponential functions were introduced in Topic 6). The function here would be f(t) = 10 t.
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 10-12 watts/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...
Question:
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?
Answer:
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.
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 energy released by the earthquake. If we use E for the response variable of energy released, and r is the explanatory variable of the earthquake's magnitude, we have another exponential function, E = 10 r.
Question:
How much stronger is an earthquake of magnitude 7.7 than one of 7.2?
Answer:
107.7 is approximately 50,118,723.4, and 107.2 is approximately
15,848,931.9.
So, 107.7 / 107.2 = 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 an earthquake measuring 7.2 on the Richter Scale.