Today students develop their understanding of the properties of logarithms. I begin by having students recall the rules for exponents. I use different parameters than most books so that students understand that the structure of expression (rather than the names) demonstrates the rules. Understanding the rules of exponents will help students understand the expansion rules for logarithms which will be developed in this lesson.
My students sometimes struggle to explain some of the rules, so I give them examples (pages 3-5) and expand the expressions with them. I give students some exponential expressions to evaluate like 2^4. I then put e^2 on the board. Students identify where e^x is on the calculator. Students give the answer and I then ask, "Does the calculator give the exact value?" After some thought students state that it is approximate because e is an irrational number. Students are given several exponential functions to evaluate. The last example allows me to remind students about fractional exponents.
I want students to connect exponential and logarithmic functions so I develop develop all concepts (graphing, evaluating, and solving) together. We'll move quickly with exponential functions, since it is review, then move on to evaluating logarithms functions. I give the students the first problem. I expect my students will use their calculator to find the answer. Then I'll say, "Okay, now explain why the answer is 5." I want to make sure that my students can to determine the value of a logarithm when the answer is a whole number especially if the base is 2 or 10.
I'll wait a minute or so for an answer to my request. If my students are not sure how to explain the answer I'll ask, "What kind of function is the inverse of a logarithm? Can you rewrite the logarithm using its inverse function?" In this case I put a box for the exponent so that the students see the problem involves finding the exponent.
We will then do several problems stating what the logarithm is saying. I will have my students rewrite the expression, and, determine the exponent that will make the exponential expression true. I use a box for the exponent at this point since we are not solving equations. I always give examples where the answer is 0 and 1. This will help the students understand the inverse properties.
The third example on page 2 allows me to introduce the term common logarithm. I ask why mathematicians call this the common logarithm. Someone usually states that our standard number system is base 10. We also look at the calculator and see that the "log" key has 10^x above the key. Again I bring up the idea of the inverse of a common logarithm is 10^x. I will introduce the ln key later in this lesson.
After working with problems where the answers are found without a calculator I give students logarithms where a calculator is used to estimate the answer. I start off by asking the students to give me an estimate of the answer. We talk about how 7 squared is 49 and 35 is less than 49. This tells us the answer is less than 2. We also discuss that the answer is over 1.5 since 35 is closer to 49 than to 7. After estimating we use the calculator to find the answer. (On the TI-84 use the alpha window.) We put the approximate answer on the board and then write the problem as an exponential equation so that students can see that we are finding the exponent.
After practicing finding logarithms of different bases, I give student a problem with a natural logarithm. "Is this function on the calculator? What is the function above the key?" We discuss that ln is called the natural logarithm and has a base of e. I write ln as a log with base e. We then use the calculator to evaluate the expression.
It is now time to connect some of the properties of exponential functions with properties of logarithmic functions. Students are given an activity to verifying logarithm properties and time to work on the activity. I move around the room to answer questions and question students about their work.
This activity gives students two proofs to analyze and use as a starting place to verify the other properties of logarithms. Learning to read mathematics is important to students. Most of my students are planning to attend college and will need to read a textbook to be successful. Students also need to communicate their reasoning and verifying properties is one type of mathematical communication.
Since students have not done many algebra proofs I scaffold the work by giving students examples to analyze and follow. If students follow the structure of the examples verifying the other properties should be straight forward. Questions 4-5 allow students to take what they have done in 1-3 and develop their own verification. This allows me to see which students are able to analyze a mathematical property, identify the structure of the property, and then use a known property to justify the validity of the property.
As class comes to an end I give students two problems to evaluate. Students are to look at the properties they have been verifying and use them to determine the answer. I will be able to assess if the students can take the abstract expression and transfer the idea to a problem.
I give students a couple of minutes to do the problems and then have students share answers on the board. Today I have all the different answers put on the board and the students explain which answer has to be correct. I ask for the property represented by the problem to justify the answer.
As a final assessment I have students turn in an exit slip with a self-evaluation of their understanding of the final two problems. Students are to state which problem they had correct before the class discussion and which problem they still need help understanding. I will use this information to lead the beginning of the next lesson.