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 3:**

**Solution:**

where

We are trying to determine the ratio of the larger magnitude *M*_{1} to the
smaller magnitude *I*_{2} or
*M*_{1}-*M*_{2}. The reason we are subtracting
the magnitudes instead of dividing them is the question asked how much
larger, not how many times larger.

Solve for *I*_{1} by multiplying both sides of the equation by *I*_{2}.

We can write
*M*_{1}-*M*_{2} as
and we can write

The larger earthquake had a magnitude 1.4 more on the Richter scale than the
smaller earthquake.

**Let's check our answer:**

Convert both of these equations to exponential equations.

**Example 4:**

**Solution:***I*_{1} represent the intensity of the stronger earthquake and *I*_{2} represent the intensity of the weaker earthquake.

What you are looking for is the ratio of the intensities:
So our task is to isolate this ratio from the above given
information using the rules of logarithms.

Convert the logarithmic equation to an exponential equation.

The stronger earthquake was 40 times as intense as the weaker earthquake.

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