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.

Example 11: Suppose that you are observing the behavior of cell duplication in a lab. In one experiment, you started with 1,000,000 cell and the cell population decreased by ten percent every minute. Write an equation with base (0.9) to determine the number of cells after t minutes.

Solution and Explanations:

First record your observations by making a table with two columns: one column for the time and one column for the number of cells. The amount of cells depends on the time. If we were to graph your findings, the points would be formed by (specific time, number of cells at the specific time).

For example at t = 0, there is are 1,000,000 cells, and the corresponding point is (0, 1,000,000). At t = 1, 10% of the cells have disappeared and there are 90% of the 1,000,000 left or 900,000 cells, and the corresponding point is (1, 90,000). At t = 2, there are 90% of the 900,000 or 810,000 cells remaining , and the corresponding point is (2, 810,000). At t = 3, there are 90% of the 810,000 or 729,000 cells remaining , and the corresponding point is (3, 729,000).

You could also say that after 1 minute the population was

You could say that after 2 minutes, the population was

After 3 minutes the population was

The population formula is therefore

where f(t) is the size of the population after t minutes, 1,000,000 is the size of the population at the start of the study (t = 0), and t is the time.

Example 12: Determine the number of cells after 10 minutes:

Solution and Explanation:

Substitute 10 for t in the equation :
Since you cannot have part of a cell, it appears that the population has declined to 348,678 cells after ten minutes.

Example 13: Determine how long it would take the population to reach a size of 10 cells.

Solution and explanation:

Here you know the number of cells at the beginning of the study and the number of cells at the end of the study, but you do not know the time. Substitute 10 for f(t) in the equation :
Divide both sides by 1,000,000:
Take the natural logarithm of both sides:
Simplify the right side of the equation using the third rule of logarithms:
Divide both sides by and simplify:

It would take a little more than 109 minutes for the population of cells to reach 10.

Example 14: Write an equation with base e that is equivalent to the equation

Solution and Explanation:

Let's start with a generic exponential equation with base e:
The f(t) represents the size of the population at time t, the t represents the time, and the a and b represent adjusters when we change the base. The value of a is also the size of the population at the start of the study (t = 0), and b is the relative growth rate with respect to the base e. Before we even calculate their values, let's see if we can determine a few things about a and b.
The starting population 1,000,000, so the value of a should be 1,000,000. The value of b should be negative, because b is the growth rate and the population is declining. Let's see if we are right.
We know that the population is 1,000,000 at time 0, so insert these numbers in the equation

We have

We now know that the value of a in the adjusted equation is 1,000,000.
Rewrite the equation with a = 1,000,000:
We know that the population after 1 minute is 900,000 cells, so insert these numbers in the equation :
Divide both sides of the equation by 1,000,000:
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 -0.10536.
Insert this value of b in the equation and the equation is simplified to
We know that the population is 810,000 after 2 minutes, so use these values to check the validity of the above equation. Substitute 2 for t in the right side of the above equation. If the answer is 810,000, then the model is correct.

The model (equation) is correct.
We know from the original equation that after 3 seconds, the population is 729,000. Let's do a second check.