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Section: A2.3.6

Going deeper: using the properties of logarithms (optional)

Solve problems using the properties of logarithms.

Students solve problems involving exponential functions with bases other than 2, $e$, or 10, or involving more than one exponential function with different bases, so that it is natural to use the property $\log(a^b) = b\log(a)$. The problems here are more complex than the ones in Section 3 and are suitable for students who are preparing for STEM majors in college.

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Tasks

WHAT: Students are given information about how a rumor spreads and write an exponential function to model the number of people who have heard the rumor on day $n$ (MP.4). It is natural to write the function with base 4 and solve the problem by taking the logarithm of both sides.

WHY: This task provides another context where most natural base to use for an exponential function is not 2, $e$, or 10, and the solution method makes use of the properties of logarithms.

2 Comparing Exponentials

WHAT: Students are given exponential functions that model the growth of two different bank accounts. One has a higher initial deposit, and one has a higher interest rate. They answer general questions about the structure of the equations and then figure out when one account will catch up with the other.

WHY: Because the two exponential functions have different bases, finding where their graphs intersect requires either the properties of logarithms or a graphical method. The task also provides an opportunity to revisit interpreting the structure of exponential functions.

3 Doubling Your Money

WHAT: Students are given the initial deposit and interest rate for a savings account. They are asked to evaluate the amount in the account after a certain amount of time, model the situation (MP.4) with an exponential (percent growth) equation, solve for how much time it will take until the amount in the account doubles.

WHY: The purpose of this task is show how logarithms can be used to find doubling times, and to explore the relationship between doubling time and the natural logarithm of 2. Students see a further glimpse into the reason that e is a natural base for a logarithm; it is the logarithm for which $\ln(1+x)\approx x$ when $x$ is small. They use this fact, and the fact that the natural logarithm of 2 is about 0.70, to derive the Rule of 70.

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1 Pounding Headache

Description

WHAT: Students are given that the half-life of acetaminophen in the blood stream is 3 hours. From there they write an exponential function modeling the amount of acetaminophen in the blood stream at any time given an initial dosage, calculate the hourly decay rate, and use logarithms to find the time at which the amount reaches a certain level given various different initial doses. They find when the drug reaches a level that is ineffective, and also investigate if it can reach dangerous levels if taken repeatedly at higher than recommended doses.

WHY: The purpose of this task is to provide a compelling context in which students explore many different aspects of exponential functions and logarithms, including building an exponential model (F-BF.A.1$^\star$, MP.4) and solving exponential equations using the properties of logarithms. Although it would be possible to write the exponential function using base 2 and solve the problem using logarithms to base 2 (F-LE.A.4$^\star$), in this task it is natural to use the hourly decay factor as the base and solve the problem by taking the logarithm of both sides and using the rule $\log(a^c)=c \log(a)$.