Discover the sum, difference, and power properties of logarithms and use them to rewrite logarithmic expressions.
Explain using logarithm properties and transformations why two logarithmic functions are equivalent.
Quick Lesson Plan
In this lesson students discover the log properties by looking at several examples and making conjectures. Your acting skills are needed in this lesson! As you monitor groups, express your curiosity and astonishment about what is happening with their values (“Whoa, that’s strange. You got the exact same thing? Why do you think that would happen?) The students really enjoy this lesson and come away with a good understanding of why the properties work (especially the power rule).
We suggest giving a warm-up reviewing exponent properties to connect ideas later in the lesson. When students see that exponents are added when powers of the same base are multiplied, they can begin to reason about why adding two logs (which represent exponents!) would cause the arguments to be multiplied.
For question 9, challenge the class to come up with as many possible logarithmic expressions that are equal to 1.806 and display them publicly. This works to spark curiosity and creativity and also assigns competence to students and their ideas.
How do you add logarithms?
How do you subtract logarithms?
When can logarithms be combined?
Can you simplify log (x+y)?
When working with exponential expressions, when do we add the exponents?
Why are these two equivalent? (Go through multiple pairs in the table in question 8)
Why is the exponent that I need to raise 10 to, to get 25, double the exponent that I need to raise 10 to to get 5?
Most of the key ideas are discovered by the students themselves during the activity, so the debrief portion should be centered around question 8. Ask students to make arguments and give reasoning for why certain statements are the same. It is easy to summarize the power property for logs as “you can pull the exponent out front”. This is a trick for remembering the property but it does not get at why the property works. It also does not highlight the relationship between the log of a number and the log of its square or cube. You can avoid this by referring to the transitive property and showing why both 2log(5) and log (5^2) are equal to log 5 + log 5, and thus equal to each other. Even better is to use something like the image below.
If you search logarithm properties on the internet you’re bound to find worksheets with hundreds of problems asking students to expand and condense logarithmic expressions. We do not take this approach here. Instead, we will focus on generating equivalent log expressions (and functions), or condensing a logarithmic expression in order to solve a logarithmic equation.
Just like in Lesson 4.4, we put emphasis on how different transformations on the parent function can produce the same graph and thus produce equivalent equations. These are not rules to be memorized (e.g. a horizontal stretch can be interpreted as a vertical shift) but should be seen from flexible manipulation of logarithmic expressions.
You may have noticed that we do not formally state the change of base formula in the QuickNotes. This is intentional. The change of base formula is often taught as a method for solving a logarithm using only the common log or the natural log. However, students know that technology can very quickly solve a logarithmic expression of any base, so this motivation falls flat. Instead, we focus on exploring relationships between logarithms of different bases such as in question 2 of the Check Your Understanding. This discovery-based model has students focusing on key characteristics of powers, as well as the exponential-logarithmic relationship, rather than memorizing a rule and applying it to rote problems.
Note about vocabulary: we do suggest using the formal word “argument” to represent the input of the logs as this will provide a consistent language for talking about all of the log properties.