APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Population Word Problems)

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation. In this section, we will review population problems.

Problem 5: Town A had a population of 10,000 in 1900 and a population of 20,000 in 1950. Find the relative growth rate and the equation that describes the population size. What was the population in 1940?

Solution and Explanations:

You could use any exponential equation of the form

where B is any positive number to solve this problem. For standardization sake, let's use the base e and the equation

The points on this graph will be (specific time, and population at that time). Let's let t = 0 at the start of our study or 1900. Therefore, our first point is (0, 10,000). The second point corresponds to the year 1950 where t = 50 (1950 - 1900). The second point is (50, 20,000).

We will use these two points to find the value of a and the value of b in the equation .

1.

Substitute the point (0, 10,000) into the equation :

2.

Substitute the value of a into the equation and we have

3.

Substitute the point (50, 20,000) into the equation and we have

4.

Divide both sides of the equation by 10,000:

5.

Solve for b by taking the natural logarithm of both sides of the equation .

6.

Simplify the right side of the equation using the third rule of logarithms:

7.

Divide both sides of the equation by 50 and simplify.

rounded to 0.0139

8.

Insert this value of b in the above equation:

The f(t) stands for the population t years after the start of the study, the 10,000 is the population at the start of the study, the 0.0139 is the relative growth rate, and t equals the number of years since the start of the study in 1900.

The equation that best fits the population trend of the city is

What exactly does best fit mean? It means that if we were to plot the population at each of the years between 1900 and 1950, the graph of the above equation would not go through all the points exactly. The graph would get as close to the actual population as possible.

The population in 1940, where t = 1940 - 1900 = 40, was

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