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 7: Suppose that you are observing the behavior of cell duplication in a lab. In one experiment, you started with 100,000 cell and observed that the cell population decreased by half every minute. Write an equation (model) with base to determine the number of cells (size of population) 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 you 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 100,000 cells, and the corresponding point is (0, 100,000). At t = 1, there are 50,000 cells, and the corresponding point is (1, 50,000). At t = 2, there are 25,000 cells, and the corresponding point is (2, 25,000). At t = 3, there are 12,500 cells, and the corresponding point is (3, 12,500).
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
Example 8: Determine the number of cells after 10 minutes:
Solution and Explanation:
Example 9: Determine how long it would take the population (number of cells) to reach 10 cells.
Solution and explanation:
It would take a little more than 13 minutes for the population of cells to reach 10.
Example 10: 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 100,000.
and the equation is simplified to
The model (equation) is correct.
The exponential equation
with base e is equivalent to the exponential equation
with base . Note that the natural log of the base is also the relative growth rate when the base is e.
By now you may have concluded that in the equation
the 100,000 represents the value at the start of the study (t = 0) and the -0.693147 represents the relative rate of growth or decline with respect to the base e.
If you would like to work another example, click on Example
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Author: Nancy Marcus