Use logarithms to solve problems
• Solve exponential equations using logarithms. • Understand the natural logarithm as a special case.
Once students understand what a logarithm is, they need ample opportunity to apply that understanding to solve problems in various contexts. All of the problems in this section should involve exponential functions with base 2, 10, or $e$ and should be solvable by reasoning directly from the definition of the logarithm, using the equivalence between $b^x = y$ and $x = \log_b y$. In particular many of them involve modeling continuous growth using an exponential function with base $e$, so this is the section where the natural logarithm is introduced. Students do not need to know the property $\log(a^b) = b \log(a)$ or solve equations using this property (“taking logs of both sides”). Note on calculating logarithms: some scientific calculators have buttons for base 10 logarithms and natural logarithms, but not base 2 logarithms. There are many online calculators that can calculate the latter, such as Desmos or the Google calculator, which is activated by typing \log_2(x)$ into the search engine.
Tasks
1
Algae Blooms
WHAT: Given an initial concentration of algae and a rule of one cell division per day, students model the growth of algae blooms MP.4 with an exponential function in base 2 F-LE.A.2. They write the results of this doubling in a table, and then find the amount of time until the cells reach a specified number which represents an algal bloom F-LE.A.4.
WHY: This context gives students a very simple exponential function to write, so the barrier to entry is low. Students can focus on solving the equation using logarithms. Note that there are two solution methods given, one using the natural logarithm which requires students to take the logarithm of both sides of an equation, and one using a logarithm to base 2, where students can reason directly from the definition of the logarithm. At this stage in the unit it is recommended that students use the second solution method.
2
Newton's Law of Cooling
WHAT: Students are given a function that models the cooling of for a cup of coffee involving an exponential function with base $e$. They are asked to interpret the meaning of the equation in the context F-LE.B.5 MP.7, and then find the time the coffee will be a certain temperature.
WHY: This task introduces students to the natural logarithm. As in the previous activity, students can reason through the solution of the exponential equation using the definition of the logarithm, and do not have to use the properties of logarithms F-LE.A.4.
3
Moore's Law and Computers
WHAT: Students are given data about hard disc capacity over the last several decades and construct an exponential function that fits the given data F-LE.A.2 MP.4. They use the model to make predictions about future storage capacity of hard drives.
WHY: Students solve a problem in a context where the solution uses a natural logarithm F-LE.A.4. This task involves doubling time, an important concept for exponential functions first introduced in the previous unit. The context in this task is the increase in hard disk storage capacity on personal computers, an important and interesting context where an exponential model has performed exceptionally well for almost 30 years. It is also a context with which many students will be familiar.
4
Snail Invasion
WHAT: Students are given two data points about the population of an invasive species. They model the situation MP.4 with an exponential function F-LE.A.2 and use logarithms to find how long it would take for the population to double F-LE.A.4. Then they use the model to answer some specific questions about the cost of eradicating the population.
WHY: The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year.