AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.13A — Exponential and Logarithmic Equations and Inequalities
Notes — Solving by Inspection and Common Bases
💡 Learning Objectives (2.13.A)
By the end of this lesson you will be able to:
- Solve exponential equations whose two sides can be rewritten with the same base
- Solve logarithmic equations by converting to exponential form
- Apply the one-to-one property of exponential and logarithmic functions
- Check solutions for extraneous values, especially in logarithmic equations
1. The One-to-One Property
Both f(x) = b^x and g(x) = log_b(x) are one-to-one — each output corresponds to exactly one input. This gives us two powerful solving tools:
- If b^M = b^N then M = N.
- If log_b(M) = log_b(N) then M = N.
These let you ‘cancel’ matching exponentials or matching logarithms on both sides of an equation.
2. Same-Base Strategy for Exponential Equations
When both sides can be rewritten with the same base, equate the exponents:
📘 Example — Solve 4^(x+1) = 8^(x−2)
- Rewrite each side as a power of 2: (2²)^(x+1) = (2³)^(x−2)
- Apply exponent rule: 2^(2x+2) = 2^(3x−6)
- Equate exponents: 2x + 2 = 3x − 6
- Solve: x = 8
3. Solving Simple Logarithmic Equations
If a logarithm equals a number, convert to exponential form:
📘 Example — Solve log₃(2x − 1) = 4
- Convert: 2x − 1 = 3⁴
- Simplify: 2x − 1 = 81
- Solve: x = 41
- Check the domain: 2(41) − 1 = 81 > 0 ✓
4. Equations with Logs on Both Sides
If both sides are a single logarithm with the same base, set the arguments equal:
📘 Example — Solve log(x + 6) = log(2x − 3)
- Same base on both sides; set arguments equal: x + 6 = 2x − 3
- Solve: x = 9
- Check domain: x + 6 = 15 > 0 ✓ and 2x − 3 = 15 > 0 ✓
5. Combining Multiple Logs Before Solving
If an equation has more than one log on a side, condense first using the properties from 2.12:
📘 Example — Solve log₂(x) + log₂(x − 2) = 3
- Condense the left side: log₂(x(x − 2)) = 3
- Convert to exponential form: x(x − 2) = 8, i.e. x² − 2x − 8 = 0
- Factor: (x − 4)(x + 2) = 0, so x = 4 or x = −2
- Check each: x = 4 ⇒ log₂(4) + log₂(2) = 2 + 1 = 3 ✓; x = −2 makes log₂(−2) undefined ✗
- Solution: x = 4 only
6. Why Extraneous Solutions Happen
When you condense logs, you replace something like ‘log(A) + log(B) = C’ with ‘log(AB) = C.’ But the second equation is defined whenever AB > 0, while the original requires both A > 0 AND B > 0. The condensed form has a larger domain, so it may produce solutions that don’t satisfy the original. Always plug your candidate back into the ORIGINAL equation.
⚠️ Common mistake
Skipping the domain check after condensing logs is the most frequent source of lost points on free-response items in this topic. ALWAYS check.
7. Simple One-Step Inequalities
The same one-to-one property gives you a way to solve simple exponential or logarithmic inequalities — but you must remember which direction the inequality goes:
- If b > 1, both b^x and log_b(x) are increasing, so the inequality direction is preserved.
- If 0 < b < 1, both functions are decreasing, so the inequality direction REVERSES.
📘 Example — Solve 2^(3x) ≥ 32
- Rewrite: 2^(3x) ≥ 2⁵
- Base 2 > 1, so direction is preserved: 3x ≥ 5
- Solve: x ≥ 5/3
8. Summary
- Use the one-to-one property to cancel matching bases or matching logs
- Convert log = number to exponential form
- Condense multiple logs before solving — then verify in the ORIGINAL equation
- Watch base size: 0 < b < 1 reverses inequalities
- Domain checks are mandatory, not optional