# Exponential and Logarithmic Properties Unit

Two students, Josh and Arobindo, investigate exponential expressions, logarithms, and logarithmic expressions. They use an exponentially-scaled number line that they created in the Exponential Functions Unit to reason about these things. If you have not watched that unit, you may want to do so before proceeding with this unit. They use the number line to explain why you can rewrite various exponential and logarithmic expressions in different forms.

##### Lesson 1: Justifying the Product Rule for Exponential Expressions

Josh and Arobindo write two equivalent exponential expressions that represent the growth of a magical beanstalk over any two consecutive time periods.  In doing so, they justify a rule for rewriting exponential expressions of a certain form.

##### Lesson 2: Justifying the Power Rule for Exponential Expressions

Josh and Arobindo write two equivalent exponential expressions that represent the growth of a magical beanstalk over several consecutive time periods of an undetermined length. In doing so, they justify a rule for rewriting exponential expressions of a certain form.

##### Lesson 3: Introducing Logarithms

The students explore the definition of a logarithm and describe a logarithm as an exponent in an exponential expression. They interpret logarithms in a moldy pizza context as the time at which a slice has a certain amount of mold.

##### Lesson 4: Interpreting a Logarithm as an Elapsed Time

The students figure out that there is another interpretation for logarithms in the moldy pizza context. They describe logarithms as the number of hours it takes for the mold to increase by a certain factor.

##### Lesson 5: Comparing Interpretations of Logarithms

The students continue to explore the two interpretations of logarithms in the moldy pizza context that they had found before, one as the hour at which the slice has a certain amount of mold and the other as the amount of time it takes for the mold to increase by a certain factor. They interpret several logarithmic expressions in both ways.

##### Lesson 6: 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.