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:
Example 13: Determine how long it would take the population to reach a size of 10 cells.
Solution and explanation:
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:
We have
We now know that the value of a in the adjusted equation is 1,000,000.
rounded to -0.10536.
The model (equation) is correct.
The equation
with base e is equivalent to the equation
with base 0.90.
If you would like to work another example, click on Example
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.
Author: Nancy Marcus