Logarithm Equation Solver
Result:
Solve Logarithmic Equations step-by-step with our comprehensive equation solver. Master different types of log equations using proven algebraic techniques and properties of logarithms.
What are Logarithmic Equations?
Logarithmic equations are equations that contain logarithms with variables in their arguments. Solving them requires understanding logarithm properties and inverse relationships with exponents.
Types of Logarithmic Equations
Simple Form
Pattern: log_b(f(x)) = c
Solution: f(x) = b^c
Example: log₂(x) = 3
Answer: x = 2³ = 8
Same Base
Pattern: log_b(f(x)) = log_b(g(x))
Solution: f(x) = g(x)
Example: log₃(x+1) = log₃(2x-1)
Answer: x+1 = 2x-1 → x = 2
Step-by-Step Solution Methods
Example: Solve log₄(x-3) = 2
Step 1: Convert to exponential form
log₄(x-3) = 2 means 4² = x-3
Step 2: Solve for x
16 = x-3
x = 19
Step 3: Check domain restrictions
x-3 > 0, so x > 3 ✓ (19 > 3)
Example: Solve log₂(x) + log₂(x-3) = 2
Step 1: Use product rule: log_a(m) + log_a(n) = log_a(mn)
log₂(x) + log₂(x-3) = log₂(x(x-3)) = log₂(x²-3x)
Step 2: Equation becomes log₂(x²-3x) = 2
Step 3: Convert to exponential form
x²-3x = 2² = 4
Step 4: Solve quadratic equation
x²-3x-4 = 0 → (x-4)(x+1) = 0
x = 4 or x = -1
Step 5: Check domain: x > 0 and x-3 > 0, so x > 3
Only x = 4 is valid
Essential Logarithm Properties
Key Properties for Solving Equations
Product Rule: log_b(mn) = log_b(m) + log_b(n)
Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
Power Rule: log_b(m^n) = n·log_b(m)
Change of Base: log_b(x) = log_c(x)/log_c(b)
Inverse Property: b^(log_b(x)) = x
One-to-One: If log_b(A) = log_b(B), then A = B
Complex Example: Mixed Logarithms
Solve: log₃(x+1) - log₃(x-2) = 1
Step 1: Apply quotient rule
log₃(x+1) - log₃(x-2) = log₃((x+1)/(x-2))
Step 2: Equation becomes
log₃((x+1)/(x-2)) = 1
Step 3: Convert to exponential form
(x+1)/(x-2) = 3¹ = 3
Step 4: Solve the rational equation
x+1 = 3(x-2)
x+1 = 3x-6
7 = 2x
x = 3.5
Step 5: Check domain restrictions
x+1 > 0 → x > -1 ✓
x-2 > 0 → x > 2 ✓
Since 3.5 > 2, the solution is valid.
Common Mistakes and How to Avoid Them
❌ Common Errors
Domain violations: Not checking x > 0 for log arguments
Extraneous solutions: Solutions that don't satisfy original equation
Property misuse: log(a+b) ≠ log(a) + log(b)
Base confusion: Mixing different bases without conversion
✅ Best Practices
Always check domain: Arguments must be positive
Verify solutions: Substitute back into original equation
Use properties correctly: Remember logarithm rules
Work systematically: Follow step-by-step approach
Applications of Logarithmic Equations
- Finance: Compound interest and investment growth
- Biology: Population growth and decay models
- Chemistry: pH calculations and reaction rates
- Physics: Radioactive decay and sound intensity
- Engineering: Signal processing and control systems
- Computer Science: Algorithm complexity analysis
Special Cases
Natural Logarithm
ln(x) = c
Solution: x = e^c
Example: ln(x) = 2
Answer: x = e² ≈ 7.39
Common Logarithm
log(x) = c
Solution: x = 10^c
Example: log(x) = 3
Answer: x = 10³ = 1000
Binary Logarithm
log₂(x) = c
Solution: x = 2^c
Example: log₂(x) = 5
Answer: x = 2⁵ = 32
Practice Problems
Basic Equation
Solve: log₅(x) = 2
Answer: x = 25
5² = 25Product Rule
Solve: log₂(x) + log₂(4) = 5
Answer: x = 8
log₂(4x) = 5 → 4x = 32Verification Strategy
Always verify your solutions by:
- Domain check: Ensure all arguments are positive
- Substitution: Replace x with your solution in the original equation
- Calculation: Verify both sides are equal
- Reasonableness: Check if the answer makes sense in context
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