Introduction
Just as logs free variables from exponents, exponents free variables from logs. But be careful—logs have strict domains, and algebra can sometimes tell lies.
Past Knowledge
Recall from Lesson 8.1 that the domain of is . This means we cannot take the log of a negative number or zero.
Today's Goal
We solve equations by exponentiating both sides. Crucially, we MUST check our answers to ensure they don't break the domain rules (extraneous solutions).
Future Success
Solving logarithmic equations allows us to work backwards from specific outcomes—like determining the concentration of ions needed to achieve a specific pH in a chemical buffer solution.
Key Concepts
The Standard Procedure
- Isolate/Condense: Get a single logarithm on one side (e.g., ).
- Exponentiate: Use the base to cancel the log (e.g., ).
- Solve for .
- CHECK: Plug the answer back into the ORIGINAL equation to ensure no negative arguments.
Worked Examples
Example 1: Single Logarithm
Solve .
Exponentiate (Base e)
.
Solve
Check: . Valid.
Example 2: Condensing First
Solve .
Condense
Use Product Rule: .
Exponentiate and Format Quadratic
.
.
Solve and Check
.
Check : . Undefined!
Answer: only.
Example 3: Logs on Both Sides (Advanced)
Solve .
One-to-One Property
If , then .
.
Solve Quadratic
.
.
.
Check Extraneous
Check : . Undefined!
Answer: only.
Common Pitfalls
Forgetting to Check
In ordinary algebra, checking is just for safety. In log algebra, checking is mandatory part of the logic. The act of exponentiating can turn undefined (ghost) solutions into real numbers. You must exorcise them.
Real-World Application
Chemistry (pH)
pH is defined as . If we know the pH is 7.4 (human blood) and need to find the ion concentration, we are solving a logarithmic equation: . This is daily work for biochemists.
Practice Quiz
Practice Quiz
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