The power of exponentials


The Richter scale for earthquakes is as follows:  if I is the intensity of an earthquake and I0 is the intensity of the shaking without an earthquake, then the magnitude R of an earthquake is defined by  

        R = log[I/I0]

The Loma Prieta quake measured 7.1 on the Richter scale and the Hokkaido quake measured 8.2.  How many times more intense was the Hokkaido quake?

Solution

Let 

        IL = The intensity of the Loma Prieta quake

and

        IH = The intensity of the Hokkaido quake

        We write

        log(IH/IL)  = log(IH/I0 / IL/I0)

        =  log(IH/I0) – log(IL/I0)

        =  8.2 – 7.1  =  1.1

By exponentiating both sides with base ten, we get

        IH/IL  = 101.1  =  12.6

We can conclude that the Hokkaido quake was more than 12 times more intense than the Loma Prieta quake.

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