Updated on August 16, 2019

#### 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 log_{3}(*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*) = log_{3}(*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 log_{3}(3^{1.5}) = 1.5, the solution to the equation log_{3}(*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.