Once again, the format of the sprints has changed.
I tell my students that any exponential or logarithmic expression involves three numbers: the base, the exponent/logarithm, and the power. When given any two of these, they should be able to determine the third, and that's the point of these sprints.
In each case they are given a logarithmic equation with one of the three numbers replaced by x. Their job is to use the other two given numbers to solve for x. Of course, this is more challenging because they have to think somewhat differently in each case - they must pay very careful attention to which element they're seeking. I won't tell them, but I will probably give them a little extra time on these.
As in the previous lesson, I want to keep the application of these concepts fresh in students minds. So at this point, I'll divide the class into a fresh set of small groups (3 to 4 students each) and hand out Exponential and Logarithmic Models 2. The students will begin collaborating while I circulate to check for understanding.
Initially, the problem requires students to formulate an exponential equation to model atmospheric pressure as a function of altitude. They are then asked to calculate the atmospheric pressure at two specific altitudes, which provide me with an easy way to check their work and them with a opportunity to get comfortable using the equation they created.
Finally, they will need to use a logarithm to answer the last question. This question is a little gruesome, but it sure gets their attention!
After about 10 minutes, I'll either call a group to the board to present their solutions via the document camera, or I'll act as the "scribe" while the students walk me through the solution together. Either way, we'll briefly review & discuss the solution, paying special attention to the creation of the equation and the use of the logarithm at the end.
Now that we have the three major properties of logarithms under our belts, it's time to summarize. Begin by asking the class once again, "So, what is a logarithm?" The answer is that a logarithm is an exponent. "Well then," I say, "when we studied exponents we generated a list of properties, so there should be a corresponding list of properties for logarithms."
The students should point out that we've already found three of them, and with that I'd begin developing the comparison table found here. (There are certainly more properties that could be added, but these are the ones I think are essential.) I will prompt the class with the properties of exponents, and ask them to come up with the corresponding logarithmic property. At this point, I will not worry about the final row in the table (the change-of-base formula).
To wrap this discussion up, and to drive home the point about domain and range, I'll ask a student to come to the board to sketch the graph of y = 10^x. Then, on the same set of axes, I'll ask for another volunteer to attempt to sketch the graph of y = log(x). We'll probably need to make some corrections to this initial sketch, so I'll call on other students to come to the board until we're satisfied. (MP 3) I'll make sure that we pay attention to the domain & range, the intercepts, and the inverse relationship between the two functions.
[This section of the lesson is beyond the scope of the CCSSM. For the reasons why I teach it, see this video.]
Now is a good time to survey our accomplishments.
"We've come to a deep understanding of logarithms and their properties. We're able to solve some problems in which they arise. But there is still one problem that we haven't fully addressed: how to calculate a logarithm."
"It's one thing to say that we can "solve" a problem like 2^x = 9 by expressing the answer as a logarithm. But it's another thing to actually express x as a number in decimal form. Our calculators are programmed to calculate logarithms with either 10 or e as a base, but what do we do if the base is not one of these numbers? Are we limited to a sort of guess-and-check with our calculator's exponent function?"
With that simple introduction, I would write an arbitrary equation on the board, such as 2^x = 9 or 17^x = 4.7.
To jump-start the students' thinking, I'll provide this one hint: We can evaluate logarithms with the base 10, so if we could convert this to an equation in base 10, then we would be able to evaluate it, too.
The question before the students should be clear: How can I convert an arbitrary exponential equation into one having a base of 10? (See this document for the argument that I have in mind.) It's worth noting that this goes beyond the CCSS for Algebra 2, but since the question usually comes up anyway, I always plan on teaching it.
I'll ask the students to consider this question in small groups, and I'll observe carefully to see what kind of progress is being made.
If they're stuck, I might offer another hint: Consider the equation 8^x = 16. We can solve this by rewriting 8 as a power of 2 like this: (2^3)^x = 16. This leads to the equation 2^(3x) = 16, which is easier to work with. Is something similar possible in this case?
Again, I'll let them consider this in their small groups. (MP 1) This hint should be explicit enough to move them in the right direction, but some will still need help seeing that they need to use a logarithm as the exponent.
My goal is to guide them toward the argument I've outlined in the resource. As soon as one of the groups figures it out, I'll have them present their "discovery" to the rest of the class and do their best to explain the reasoning behind it. (MP 3) Once everyone has had their questions answered, I'll be sure to provide a formal statement of the change-of-base formula for their notes.
For a final send-off, I'll ask everyone to use this new formula to solve another problem I'll make up on the spot. Their ticket out the door will be to express the answer as a ratio of logs to the base 10 and also use their calculator to evaluate it.