Algebra II: Solving Logarithmic Equations

Objectives

• Students will numerically approximate solutions to logarithmic equations
• Students will graphically determine exact solutions to logarithmic equations using the functions f(x) = log_a(x) and f^-1(x) = a^x and the composition
• Students will find exact solutions to exponential equations using algebraic techniques that employ the relationship f ° f^-1(x) = x

Vocabulary

• Exponential functions and equations
• Logarithmic functions and equations
• Inverse functions
• Composition of functions

About the Lesson

This lesson involves numeric, graphical, and algebraic solutions to the equation log₃(x) = 1.5. Students will:

• Analyze numeric patterns, predict an approximate solution, and evaluate predictions in a spreadsheet.
• Consider the graphs of both f(x) = log₃(x) and f^-1(x) =3^x to determine that f(x) = 1.5 precisely when f^-1(1.5) = x.
• Use the compositional relationship of log_s(3^x) = x to solve the equation. Since log₃(3^1.5) = 1.5, the solution to the equation log₃(x) = 1.5 is x = 3^1.5.
• Consider composition in the opposite order, solving the equation algebraically by employing 3^log₃(x) = x
• Use these techniques to solve similar equations.