Justifying the Product Rule for Logarithmic Expressions
Josh and Arobindo write two equivalent exponential expressions that represent the time it takes for the amount of mold to increase by one factor, and then by another. In doing so, they justify a rule for rewriting logarithmic expressions of a certain form.
Episode 1: Making Sense
The students model on their number line the mold growth on the pizza over two consecutive time periods. They then calculate the total elapsed time over the two periods of time.
Episode 2: Exploring
Josh and Arobindo consider another student’s equation that is meant to describe the mold growth over two consecutive time periods. They decide the student has written an incorrect equation and propose a new equation. They explain why the two logarithmic expressions in their equation are equivalent.
Episode 3: Repeating Your Reasoning
Arobindo and Josh fill in the blanks of the equation log3 (___) + log3 (___) = log3 (1,000) to make the equation true. They justify their answer.
Episode 4: Reflecting
The students find a logarithmic expression that is equivalent to log3 (x) + log3 (y). They explain what their equation means in terms of the moldy food context and justify their proposed relationship using a number line.