APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**EARTHQUAKE WORD PROBLEMS:**

As with any word problem, the trick is convert a narrative statement
or question to a mathematical statement.

Before we start, let's talk about earthquakes and how we measure their
intensity.

In 1935 Charles Richter defined the magnitude of an earthquake to be

where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a ''standard earthquake'' (whose amplitude is 1 micron =10

The magnitude of a standard earthquake is

Richter studied many earthquakes that occurred between 1900 and 1950. The
largest had magnitude of 8.9 on the Richter scale, and the smallest had
magnitude 0. This corresponds to a ratio of intensities of 800,000,000, so
the Richter scale provides more manageable numbers to work with.

Each number increase on the Richter scale indicates an intensity ten times
stronger. For example, an earthquake of magnitude 6 is ten times stronger
than an earthquake of magnitude 5. An earthquake of magnitude 7 is
times strong than an earthquake of magnitude 5. An earthquake of
magnitude 8 is
times stronger than an earthquake
of magnitude 5.

**Example 5:**

**Solution:**

Convert the ''The 1976 Tangshan earthquake was 1.26 as intense.'' to an
equivalent mathematical equation.

where
*I*_{Mexico} is the intensity of the Mexico City earthquake and
*I*_{Tangshon} is the intensity of the Tangshon earthquake.

We are trying to determine the magnitude of the Tangshon earthquake.

**Let's check our answer:**

Convert both of these equations to exponential equations.

**Example 6:**

**Solution:***M*_{L} represent the intensity the
earthquake with the greater intensity and *M*_{S} represent the
earthquake with the smaller intensity.

What you are looking for is the ratio of the intensities:
*M*_{L}-*M*_{S}. So
our task is to isolate this expression from the above given information
using the rules of logarithms.

The difference between the magnitudes of the two earthquakes is 2.11 on the
Richter scale.

**Let's check the answer.**

It will not check exactly because we rounded the 2.11, but it is
sufficiently close to check our answer.

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