Cogito
Integrated Math 3 · Chapter 4 · Lesson 3
Solving Exponential and Logarithmic Equations
Getting the unknown out of the exponent.
12 problems · about 22 minutes · F-LE.A.4, A-REI.D.11
What this lesson teaches
The student solves exponential and logarithmic equations and checks the domain.
- Take logs when the unknown is in the exponent.
- Matching bases lets you equate exponents directly.
- Combine logs before converting, then check every argument is positive.
Warm Up
Straightforward practice. Get the method working first.
5 problems4ˣ = 64. What is x?
Answer 3
Why 3.
Why must log equation solutions be checked?
Answer Combining logs can widen the domain, admitting invalid values.
Why The combined form permits values the original did not.
2ˣ = 64. What is x?
Answer 6
Why 2⁶ = 64.
3ˣ = 81. What is x?
Answer 4
Why 3⁴ = 81.
log base 3 of x = 2. What is x?
Answer 9
Why 3².
Build It Up
The same ideas with more to keep track of.
3 problemslog base 5 of x = 3. What is x?
Answer 125
Why 5³.
5ˣ = 1. What is x?
Answer 0
Why Anything to the zero.
A candidate makes a log argument negative. Is it valid? 1 yes, 0 no.
Answer 0
Why Reject it.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the solving process for a log equation in order.
Answer 1. Use the laws to combine into a single logarithm. 2. Rewrite in exponential form. 3. Solve the resulting equation. 4. Reject any candidate that makes an argument non-positive.
Why Combining comes before converting.
10ˣ = 1000. What is x?
Answer 3
Why 10³.
The Same Base: 2ˣ = 32. What is x?
Answer 5
Why 5.
The Log Equation: log base 2 of x = 4. What is x?
Answer 16
Why 16.