M5.2.4 For the adult
Equations with powers
How do we solve an equation where the unknown is in a power, such as 3ˣ = 20, when trying numbers is slow?
What this chapter teaches
To solve aˣ = b, take logs of both sides and bring the power down: x = log b ÷ log a. A disguised quadratic such as 4ˣ − 5(2ˣ) + 4 = 0 is solved with u = 2ˣ, keeping only roots with u above 0. In an equation with logs, every answer must be checked so that no number inside a log is zero or negative.
Three questions to ask
There is no one right wording. Listen for the idea.
- Why is x = 20 ÷ 3 not the answer to 3ˣ = 20? How could you test it?
- Why can the equation 2ˣ = −8 never have a solution?
- What check should you do after solving an equation with logs in it, and why?
After the chapter, the learner can
- Solve 3ˣ = 20 by taking logs
- Solve 4ˣ − 5(2ˣ) + 4 = 0 as a quadratic in 2ˣ
What they copy into their notebook
aˣ = b: take logs, x = log b ÷ log a. Disguised quadratic: let u = aˣ, keep u > 0. Log equations: one log, index form, reject answers that make a number inside a log 0 or less.