Solve exponential and logarithmic equations by rewriting them in equivalent forms using properties.
Identify and exclude extraneous solutions to exponential and logarithmic equations.
Write equations for inverse functions by applying inverse operations.
Quick Lesson Plan
Today’s experience presents a learning progression from very simple to very complex logarithmic equations. First, students start by solving a quadratic equation to remind themselves of the key skills of combining like terms and inverse operations. Next, students reason about missing exponents and missing inputs of log equations and discover the value of rewriting an equation in its alternate form to isolate the question mark and make sense of the equation. In question 5, students must now complete at least one extra step to isolate their variable. In parts a and b, the extra step is to complete an inverse operation, in part c, the extra steps are to apply a log property and then complete an inverse operation.
Students were very successful on question 6 and again were asked to think about the idea of equivalence. In part d, they had to also apply a log property.
Question 7 is by far the most challenging, as students have to apply everything they’ve learned so far. If some groups don’t get to the end of this, that’s okay! Feel free to start the debrief when groups are still in a state of uncertainty about question 7, as this will increase engagement and heighten their desire for new knowledge. A key part of debriefing this question is getting students to notice that even though algebraically there are two solutions (x=3 and x=-2), not both values of x are defined in the original equation. Solving a logarithmic equation requires checking to make sure that your final answers are actually defined inputs of the original equation! If not, we consider those solutions extraneous, and we remove them from the solution set.
How did you determine which value(s) of x make this equation true?
What’s the difference between evaluating and solving?
Thinking about the function y=x^2, if I’m looking at a point where the y-value is 9, what must the x-value be? Is there a definitive answer? Why or why not?
How is a quadratic function different from an exponential and logarithmic function in terms of its graph and number of solutions?
Where do the functions y=4^x and y=4^11 intersect? Is there more than one place?
Where do the functions y=log base 4 of x and y=log base 4 of 10 intersect? Is there more than one place?
How do I “combine like terms” with logarithms?
What is the domain of a logarithmic function?
The goal of today’s lesson is to gather strategies for solving logarithmic and exponential equations, more so than to define a procedure for solving those equations. In the debrief, ask students to articulate why certain strategies were helpful and what the purpose of using that strategy is.
While question 6 was easy for students to complete, the tie-in to the one-to-one property is more challenging, as students don’t readily see both sides of the equations as outputs of a function. To provide a non-example of the one-to-one property, ask students to think about y=x^2, a function that is NOT one-to-one. Ask them if knowing that the output is 9 would let them know for sure what the input is. When students say that the input could have been 3 or -3, ask them if this same scenario could occur with exponential or logarithmic functions. “Is it possible that two different inputs would have produced the same output?”
To help students apply the one-to-one property, have them identify what was special or different about these equations from the equations in question 5. Students should be able to articulate that they saw exponential or logarithmic expressions with the same base on both sides of the equation. This is a defining characteristic that will help students on questions like 2b on the Check Your Understanding, where they should recognize that 8 can be written as 2^3 so both sides of the equation show the same base and thus the exponents can be set equal to each other.
One final idea in today’s lesson is that finding an inverse function uses the exact same process as solving an exponential and logarithmic equation! Why? Because we learned in Lesson 5.2 that finding an inverse means isolating the other variable, so that what was the independent variable can now be the dependent variable.