Today I introduce my students to logarithmic functions with bases other than 10 by starting with a base 10 log on the board and challenging them to describe what each number represents. Generally they will use exponential functions to help with the description. (MP6) Once we've identified the expression components, I post another logarithmic function, this one with a base of 2. I again ask for volunteers to describe the components and help with directed questions if necessary. For example I might ask "What does that component represent in a base 10 logarithm?" I add one more logarithmic function to the board, this one with a base of e. I write this one using both log and using ln. By now my students know what we're looking for and can fairly quickly describe the components, but may stumble over the "ln" notation. I ask what they think the "ln" might represent and usually have at least one student who suggests that it means log base e. (MP7) If not, I point out that log base 10 doesn't always include the "10" when writing it out, in fact their calculator has a log key which represents log base 10, then ask again what "ln" might represent.
I begin this part of the lesson with brief whole class instruction on logarithmic functions and graphing calculators. I walk through examples of logarithmic functions on the board, then have my students see what they can work out using their calculators. (MP5) Some of my students are very comfortable with their calculators but for others these are still relatively new tools that can be a bit intimidating so for those students I provide direct instruction if necessary. Most students pick up on the log and ln keys fairly quickly, but only a few figure out how to find logs with bases other than 10 or e, so I demonstrate that process for the class so they can check their rewritten functions. I use fist-to-five to check for understanding then tell my students they will have an opportunity for individual practice so they can gain confidence and skill with logarithmic functions. I distribute the practice worksheet and remind them to show their work. (MP1) While they work I walk around offering encouragement and redirection as needed. When everyone is done, I have them self-check their papers so they have immediate feedback and an opportunity to ask questions while their work is still fresh in their minds.
After answering any questions, I tell my students that for the next part of the lesson they get to try to stump each other by creating their own logarithmic functions. (MP1, MP2) My video explains why I think this activity is valuable. I distribute the Logarithm Game Rules and a blank piece of paper to each student then wait until everyone has read it and responded to the direction to get a mini-whiteboard and marker. When everyone is again seated I know they're ready to begin and give the direction to start. While they're competing I walk around observing and serving as final arbiter if necessary. I prefer to have my students sort out who is correct by asking questions like "How could you check that answer?" or "What components represent what in this function?" I call time when there are only five minutes left in class and congratulate my students on a good challenge.
To close this lesson I give each student a notecard and ask them to write one logarithmic problem they think will stump me as their ticket-out-the-door. (MP2) I tell them they have to record the problem and their rewrite in exponential form in their notes so we can check it tomorrow. This gives them a chance to show what they know and lets me know which students are still struggling. By asking them to try to stump me, I throw down a gauntlet most students can't resist, so they push themselves a bit further than they otherwise might to come up with a really tough problem...a great way to help them stretch a bit!