Lesson 5.17
Solving Logarithmic Equations
To solve equations containing logs, condense to a single log, then convert to exponential form. Always check for extraneous solutions — you can't take the log of a negative number or zero.
Introduction
Log equations come in two types: those with a single log (convert to exponential) and those with multiple logs (condense first, then convert). Either way, the final step is checking that no argument becomes negative or zero.
Past Knowledge
Converting forms (5.7), expanding/condensing (5.14), domain restrictions.
Today's Goal
Solve log equations and reject extraneous solutions.
Future Success
Application problems in 5.18 require solving both exponential and log equations.
Key Concepts
Type 1: Single Log = Number
Convert directly to exponential form.
Type 2: Log = Log
If the logs are equal (same base), the arguments are equal.
⚠️ Always Check: Arguments Must Be Positive
After solving, substitute back into every log argument. If any argument ≤ 0, that solution is extraneous — reject it.
Worked Examples
Example 1: Single Log
BasicSolve .
Convert to exponential
Solve and check
Check: ✓
Example 2: Two Logs Combined
IntermediateSolve .
Product property to condense
Convert to exponential
Solve the quadratic
Candidates: and
Check both
x = 5
✓ and ✓ — both positive
x = −2
is undefined → Extraneous!
Only
Example 3: Log = Log
AdvancedSolve .
Same base → set arguments equal
Check: both arguments positive?
✓ and ✓
Common Pitfalls
Skipping the Domain Check
Log equations almost always produce extraneous solutions. Never skip checking that every log argument is positive.
Condensing Before Isolating
Make sure all log terms are on one side and non-log terms on the other before condensing.
Real-Life Applications
pH equations like are solved for using these techniques. Sound engineering, seismology, and chemistry all require solving log equations regularly.
Practice Quiz
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