Logarithms

Before this unit, students can construct exponential functions given a graph, table, or description, and interpret exponential functions expressed in different forms in terms of a context. They can graph exponential functions and note key features such as end behavior, asymptotes, and intercepts. They recognize real-world contexts that can be modeled by exponential functions, such as savings accounts and population growth, and they construct functions to model them. For example, they might describe the amount in a savings account with a $1000 initial balance that earns 10% a year compounded monthly using the function $A(t) = 1000(1.0083)^{12t}$. They can solve problems involving exponential equations like $20,000 = 1000(1.0083)^{12t}$ graphically, but not algebraically.

In this unit, students understand the logarithm defined operationally as the inverse of exponentiation, and have opportunities to practice interpreting logarithm notation and evaluating logarithms. They use logarithms to solve for an unknown exponent in situations modeled by exponential functions. These functions include ones expressed with base e, necessitating the introduction of the natural logarithm. Students graph logarithmic functions along with the exponential functions that are their inverses, developing an understanding of logarithmic functions as the inverses of exponential functions.

The Common Core State Standards do not explicitly require students to know and use the properties of logarithms. Student intending to pursue STEM careers in college should go deeper and learn these topics. Two optional sections at the end of the unit develop and apply the properties of logarithms.

After this unit, logarithms become a natural part of the toolkit in working with situations modeled by exponential functions. Applications abound in calculus, engineering, and the sciences. There are many sets of data that reveal their structure when plotted on logarithmic scales.