APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Amortization Word Problems)

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

Alright, this is where we start to get serious.

Example 17: A city in Texas had a population of 75,000 in 1970 and a population of 200,000 in 1995. The growth between the years 1970 and 1995 followed an exponential pattern. Find a model (equation) that estimates the population at any time during those years. From this model, estimate the population for the year 2010.

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 t = 0 at the start of our study (in our case the year 1970). Therefore, our first point is (0, 75,000). The second point corresponds to the year 1995 where t = 25=(1995 - 1970). The second point is (25, 200,000).

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

• Substitute the point (0, 75,000) into the equation :

  

• Substitute the value of a into the equation

  

  and we have

  

• Substitute the point (25, 200,000) into the equation

  

  and we have

  

• Divide both sides of the equation by 75,000:

  

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

  

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

  

• Simplify the left side of the equation:

  

  rounded to 0.98083.

• 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 75,000 is the population at the start of the study, 0.98083 is called the relative growth rate, and t equals the number of years since the start of the study in 1970.

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

What exactly does best fit mean? It means that if we were to plot the population at each of the years between 1970 and 1995, 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.

We know nothing of the future; therefore, when you are asked to estimate a population for the future, you are working with an educated guess. For example, we asked you to estimate the population in the year 2010. I don't have a crystal ball, do you? How then do we make this guess?

We first assume that everything is going to continue for the next 15 years as it has for the past 25 years. That is a pretty tall order. It is not too much of a stretch to predict for the few years following 1995, but all the way to 2010, not so easy. We have to assume that there are no wars, no depressions, no famine, no one found gold or oil outside the city, no plague wiped out the city, and so forth. You get the picture.

Be very careful when making predictions far into the future. If you make these predictions with a great deal of accuracy, I want you for my stock broker. (Kidding)

With that said, and all the above assumptions made, to make the prediction for the year 2010, simply substitute 2010 - 1970 = 40 for t in the model we derived.

If everything continues for the next 15 years as it has for the past 25 years, the city will have a population of 2,942,490 in the year 2010.

If you would like to work another example, click on Example

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