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 4: Suppose that you are observing the behavior of cell duplication in a lab. In one experiment, you start with one cell and the cell population is tripling every minute. Write an equation with base 3 to determine the number of cells after one hour.
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 number of cells (population size) 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 1 cell, and the corresponding point is (0, 1). At t = 1, there are 3 cells, and the corresponding point is (1, 3). At t = 2, there are 9 cells, and the corresponding point is (2, 9). At t = 3, there are 27 cells, and the corresponding point is (3, 27).
You can see that the relationship between the two parts of the point is exponential where the exponent is the time. Therefore, we say that the equation that reflects the (number) size of the population at time t is
Let's check it by estimating the population after 10 minutes with the formula and with the table. By the formula,
By the table, after 10 minutes the population is
Therefore the formula is
Determine the number of cells after one hour:
Solution and Explanation:
Example 5: Determine how long it would take the number of cells (population) to reach 100,000 cells.
Solution and explanation:
It would take a little more than 10 minutes for the number (population) of cells to reach 100,000.
Example 6: Write an equation with base 10 that is equivalent to the equation .
Solution and Explanation:
You now know that the value of a in the adjusted equation is 1. You really already knew this since the number of cells at time 0 was 1.
which in turn can be rewritten as
rounded to 0.4771.
and the equation is simplified to
rounded to 27. Remember the check won't be exact because we rounded 0.47712154721 to 0.477.
with base 10 is equivalent to the equation
with base 3.
By now you may have concluded that in the equation , the value of a is the size of the population at time zero and b is the relative growth rate with respect to the base e. If b is positive, the population is growing. If b is negative, the population is declining.
If you would like to work another example, click on Example
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Author: Nancy Marcus