Unit 5
Exponential Functions and Equations
Unit Narrative
In this unit, students explore exponential and logarithmic functions. The unit begins with students recalling geometric sequences and drawing a connection to the growth or decay of values by a constant growth factor. This leads to expressing exponential relationships using functions of the form
Students use different rational inputs, including negative values and values between integers, to better understand exponential functions and their meaning in various contexts. This includes an exploration of growth factors over intervals of different lengths. For example, the same population growth can be described using a growth factor per decade or a different growth factor per year.
The exploration of variables used as an exponent leads to a need to solve equations for the variable and to the introduction of logarithms. Students are presented with traditional logarithm patterns and asked to find patterns and relationships, which leads them to discover that logarithms represent a way to rewrite exponential equations. That is,
The constant
Finally, students solve exponential and logarithmic equations using graphical and algebraic methods and then interpret the solutions in context. The logarithms in this unit are primarily focused on the bases 10, 2, and
Note that, throughout the unit, the bases for both exponential expressions and logarithms are assumed to be positive.
Coordinate plane, horizontal, time since study began, months, 0 to 25 by 5, vertical, ant populations, thousands, 0 to 16 by 4. Dotted lines connect 20 comma 15 million to corresponding spots on the x and y axis. Line c begins at 0 comma 8 point 1. Line r begins at 0 comma 5 point 4. The lines meet at approximately 20 comma 15 million.