In this lesson, students continue the work of connecting exponential functions and logarithms, using them to compare functions and solve problems.
In both activities here, students analyze functions graphically and algebraically. They recognize the intersection of two graphs as the point when both functions have the same value and interpret that point in context (MP2). They also practice writing equations and solving them using a logarithm. Students see that when graphs of two functions \(f\) and \(g\)intersect, the \(x\)coordinate of the intersection is a solution to the equation \(f(x) = g(x)\).
Note that students have not yet encountered a function with base \(e\) and a negative exponent (which is commonly used in many realistic decay contexts). Though the meaning of expressions of the form \(e^{rt}\) when \(r\) is negative is not explored in this unit, consider sharing with students that when an exponential decay is expressed with a function of the form \(e^{rt}\), the exponent \(r\) takes a negative value.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 16.1 Warmup: Two Bank Accounts (5 minutes)

16.2 Activity: Bacteria in Different Conditions (20 minutes)
 Includes "Are you Ready for More?" extension problem
 16.3 Activity: Populations of Two Countries (10 minutes)
 Lesson Synthesis
 16.4 Cooldown: Reaching 40 Thousand (5 minutes)
Learning goals:
 Calculate the coordinates of the point of intersection of two exponential graphs using logarithms.
 Interpret the intersection of the graphs of two exponential functions in context.
Learning goals (student facing):
 Let’s compare exponential functions by studying their graphs.
Learning targets (student facing):
 I can calculate where two exponential graphs meet using logarithms.
 I can interpret the intersection of the graphs of two exponential functions in context.
Required materials:
 Scientific calculators
Standards:
 This lesson builds on the standards: CCSS.HSAREI.D.11CCSS.HSFLE.A.4MS.AREI.11MS.FLE.4MO.A2.REI.B.3MO.A2.SSE.A.2MO.A2.SSE.A.3
 This lesson builds towards the standards: CCSS.HSAREI.D.11CCSS.HSFIF.A.2MS.AREI.11MS.FIF.2MO.A2.REI.B.3MO.A1.IF.A.2
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