APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**DECAY WORD PROBLEMS:**

To solve an exponential or logarithmic word problem, convert the
narrative to an equation and solve the equation.

**Example 4:** Alpino, New Mexico had a population of 20,000 in
1900 and a population of 5,000 in 1995. The population decline followed an
exponential model. What was the population in 1950? If we can assume that
the model holds into the future, in what year will the city have less than
10 people? Use the base 10 to find your answers.

**Solution and Explanation:**

Let's us form the equation with base 10:

Let's call 1900 time 0. The population in 1900 was 20,000.

The equation is now

In 1995 when *t*=95, the population was 5,000. Another way of saying this is

Take the common log of both sides of the equation.

The equation can be written as

In 1950, *t*=50. Just substitute 50 for t in the equation.

In the year 1950 the population was 9,642.

In what year will the population be less than 10 people? Substitute 10 for

Take the common log of both sides of the equation.

In the year 1900 + 521 or in the year 2421, the population will be less than
10.

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