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 = 25
Product Rule

Solve: log₂(x) + log₂(4) = 5

Answer: x = 8

log₂(4x) = 5 → 4x = 32

Verification Strategy

Always verify your solutions by:

  1. Domain check: Ensure all arguments are positive
  2. Substitution: Replace x with your solution in the original equation
  3. Calculation: Verify both sides are equal
  4. Reasonableness: Check if the answer makes sense in context

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