4.3: Logarithms & Exponential Models
Contents
Objectives:
- Given an exponential model and an output, use logarithms to solve for input.
- Compute doubling time and half-life
- Convert between the forms $f(t)=a(b)^t$ and $f(t)=ae^{kt}$
Lesson Guide
Warm-Up
Have students do Problem 1.
Guide students through problem 2
Given an exponential model and an output, use logarithms to solve for input.
Introduce an exponential model in the context of a word problem.
-Example:
Use an exponential model where the inputs are years and the outputs are sizes of a population.
Ask students to help determine how given a desired population, one can find the year in which that population is achieved. Connect this idea back with the notion of logarithms as inverse functions of exponentials. Emphasize the importance of writing the final answer in a complete sentence with correct units.
While solving, do every step algebraically and THEN put the final answer into a calculator. Tell students that if they enter each step into their calculator and round that answer, the final answer may be way off. Do not round until the final step, if at all.
Have students do Problems 3 and 4.
Compute doubling time and half-life
Do some examples given as word problems that include the phrases "doubling time" and "half-life"
-Example:
-Example:
Have students do part (a) of Problem 6
Convert between the forms $f(t)=a(b)^t$ and $f(t)=ae^{kt}$.
Explain to students that given a growth factor $b$, we can always rewrite $b$ as $e^k$ for some $k$ using logarithms. In doing so, we can convert an exponential model the form $f(t)=a(b)^t$, which clearly exhibits the annual growth rate, to an equivalent model $f(t)=ae^{kt},$ which clearly exhibits the continuous growth rate. Do an example of this.
-Example:
Conversely, given a continuous growth rate $k$, we can always find a growth factor $b$ such that $b=e^k$ by simply evaluating $e^k.$ Then, we can determine what the effective annual growth rate, $b-1$, is. In doing so, we can convert an exponential model $f(t)=ae^{kt}$ to an equivalent model $f(t)=a(b)^t.$ Do an example of this.
-Example:
Students often struggle with this concept. Emphasize to students that the two exponential expressions model identical populations, but the format $f(t)=ae^{kt}$ clearly shows the continuous growth rate while $f(t)=a(b)^t$ clearly shows the effective annual growth rate.
Have students do Problems 7-8.
Throughout class, pause and have students present their solutions. Try to ensure that all students feel comfortable with the basic concept of the relationship between logarithms and exponential models.
Do the Focus Problem.
Comments
I found it hard to motivate why we would want to convert between between annual and continuous models. I gathered the following reasons from a conversation with Nathan. Feel free to modify or add to these.
- In finance, biology, or chemistry, different models may be used in different scenarios. The ability to convert between the two models could help them better compare models that arise if they appear in different forms.
- It's valuable for students to be able to tell the differences between continuous versus annual models so that they can understand how and why the growth factors/rates are different. For example, since continuous growth models grow faster, the continuous rate should always be expected to be smaller than the annual rate.
