Logarithm Properties (Lesson 3.6)

Unit 3 - Day 7

​Learning Objectives​
  • Discover the sum, difference, and power properties of logarithms by using inductive reasoning

  • Use properties of logarithms to expand or condense expressions

  • Identify and write equivalent logarithmic expression

Quick Lesson Plan
Activity: Puzzling with Properties



Lesson Handout

Answer Key

Experience First

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.

Formalize Later

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.


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.


Although the skills of expanding and condensing logarithmic expressions might seem somewhat rote, this is an important sub-skill that students will use to solve logarithmic equations in the subsequent lessons. It is also a great opportunity to emphasize and review the idea of equivalence. When giving an assessment, we rarely ask students to expand and condense expressions, but we will ask them to identify or generate equivalent log expressions, or to apply the skill of expanding and condensing when solving a logarithmic equation.

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