Ask students to write an equivalent form of log419. Students will likely come up with log 19/log 4. Ask if they can think of another one and see if they can come up with ln 19/ ln 4. Now see if students can generalize this relationship using variables. If they can, then they have recalled one property of logarithms that they probably already know. Let them know that they will be reviewing other properties of logarithms today during our Scavenger Hunt.
We are going to use a Scavenger Hunt to review these properties. In the video below I explain what a Scavenger Hunt is and explain the process.
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The directions will be simple for students: start at any of the posters and find the equivalent form of the logarithmic expression. If students are not sure, they can use calculators or plug in values to see what is equivalent. Look around at the bottom of the other posters to find the equivalent form. Move on to the next poster and repeat the process. Students should go to every poster exactly once and should end up where they started. It is really important that they keep a record of the equivalent expressions so they can reference them later. They should also write down the order of the posters they went to so that they can verify that their answers are correct.
After students work on the Scavenger Hunt, read the correct order of the letters out loud so that students can check their work. Pick one or two equivalent expressions and have a student explain how they know that they are equivalent. The nice thing about this activity is that students could guess the properties even if they do not remember them. For example, they may have used their calculator to deduce that log 20 = log 4 + log 5, but they will probably be able to guess that you multiply the 4 and 5 to get the 20.
Next, have students generalize these properties. Tell students that there were three different properties (multiplication, division, and exponents) that they worked with and challenge them to write the properties using variables. Give them a few minutes to work on this. Have students share their properties and see if the class agrees with them.
You do not have to go through a formal proof for every property, but it is nice to speak about why they are true. When discussing log (ab) = log a + log b, for example, you can ask students why that is true. Get them to see that a logarithm is just a weird way of writing an exponent, and when exponential expressions are multiplied (like a^2 times a^7), you simply add the exponents to get a^9. They will be able to see the correlation to the properties that we looked at. If you have time, you may want to go through a formal proof of one of them just to formalize what we have been working on.
Students may wonder why we even care to write these logarithmic expressions in different ways. In reality, there is not really a need until they get to calculus and want to start finding derivatives and antiderivatives of these expressions – separating the expressions into terms of the form alogb simplifies things greatly. I’ll often tell students this so at least they know they will be using these properties in the future.
Finally, an assignment will summarize the work that we did and give them some more practice.