If students are unsure what  means, because it’s their first time working with logs as functions, consider saying:

• “Tell me more about what you notice in the equation .”

• “How could rewriting the equation in exponential form help you to better understand it?”

The purpose of the discussion is for students to become familiar with the graph of a logarithmic function. Display the graph of .

Focus the discussion on the graph of the function and students’ observations of its features. Ask them to describe how the graph is different from graphs representing exponential, linear, quadratic, or other functions they have seen.

Introduce as a logarithmic function. In this case, the base of the logarithm is 2, which tells us it is related to a quantity that changes by a factor of 2. Here, it is used to find the time it takes a population that doubles every hour to reach a certain number. Discuss with students:

• “What does the input of this function mean in this situation?” (the population of bacteria)
• “What does the output mean?” (the time, in hours, that it takes to reach a certain population size)
• “What does the expression mean? What is its value?” (the number of hours it takes the colony to reach 24 thousand bacteria, which is about 4.6 hours)
• “How did you find the time it takes the colony to reach 72 thousand bacteria?” (I looked on the graph I sketched to estimate to find where a vertical line at  would meet the curve. It looks to be a little more than 6, so I estimated 6.2 hours.)

Point out that, like graphs of other functions, log graphs can be used to answer questions about the relationship, though not always precisely. We can, however, use the equation defining a log function to solve problems with exact answers.

If time allows, conclude the discussion by asking, “About how many hours will it take for the population to reach 1 million?” (It would take about 10 hours because one million bacteria equals 1,000 thousand bacteria and . So,  thousand is a little more than 1 million. Or, using a base-2 log table or a calculator with function, , which is about 10.)

Make sure students see that this bacteria population is growing by a factor of 10 every day. Highlight that here the input (population) increases tenfold when the output (days) increases by 1. Relate this observation to the meaning of base-10 logarithm.

Focus the discussion on students’ observations and interpretations of the graph, and how the graphs representing and compare.

Invite previously identified students who use a graph to answer the last question to share their approach. If no one took this path, ask students to consider if it is a helpful strategy. Students may be able to easily see on the graph the input value that gives an output of 4, but to use the graph to find out what input value gives an output of 5 requires looking into very large input values. A key takeaway here is that the input has to increase by a factor of 10 for the output to increase by 1.

Tell students that logarithmic functions do increase forever and can reach any positive output for large enough inputs. For example, there is a point on the graph of at . It may not make sense in real situations like this to use such large inputs, but it is important to know that the graph does not have a maximum value.