📊 Logarithm Equations & Inequalities

Master logarithmic equation solving with domain awareness and systematic approaches.

🎯 Recognition Cues

Look for these patterns:

  • Single logarithm: $\log_a x = b$
  • Multiple logarithms: $\log x + \log(x+1) = 2$
  • Logarithmic inequalities: $\log_2(x-1) > 3$
  • Change of base: $\log_3 7 = \frac{\log 7}{\log 3}$
  • Exponential-logarithmic: $2^x = 5$

📋 Template Solution (4 Steps)

  1. Check domain - All arguments must be positive
  2. Simplify - Use logarithm laws to combine terms
  3. Solve - Convert to exponential form or use properties
  4. Verify - Check solutions satisfy domain and original equation

🔍 Common Patterns

Pattern 1: Single Logarithm

Template: $\log_a x = b$ → $x = a^b$

Example: Solve $\log_2(x-1) = 3$

Solution:

  1. Domain: $x - 1 > 0$ → $x > 1$
  2. Convert: $x - 1 = 2^3 = 8$
  3. Solve: $x = 9$
  4. Check: $x = 9 > 1$ ✓ and $\log_2(9-1) = \log_2 8 = 3$ ✓

Pattern 2: Multiple Logarithms

Template: Use laws to combine: $\log_a x + \log_a y = \log_a(xy)$

Example: Solve $\log x + \log(x+1) = 2$

Solution:

  1. Domain: $x > 0$ and $x + 1 > 0$ → $x > 0$
  2. Combine: $\log(x(x+1)) = 2$
  3. Convert: $x(x+1) = 10^2 = 100$
  4. Solve: $x^2 + x - 100 = 0$ → $x = \frac{-1 \pm \sqrt{401}}{2}$
  5. Check domain: Only $x = \frac{-1 + \sqrt{401}}{2} > 0$ ✓

Pattern 3: Logarithmic Inequalities

Template: Consider base: if $a > 1$, direction preserved; if $0 < a < 1$, direction reversed

Example: Solve $\log_2(x-1) > 3$

Solution:

  1. Domain: $x - 1 > 0$ → $x > 1$
  2. Base 2 > 1: Direction preserved
  3. Convert: $x - 1 > 2^3 = 8$
  4. Solve: $x > 9$
  5. Combine: $x > 9$ (satisfies domain)

Pattern 4: Change of Base

Template: $\log_a x = \frac{\log_b x}{\log_b a}$

Example: Evaluate $\log_3 7$ using base 10

Solution:

  • $\log_3 7 = \frac{\log 7}{\log 3} \approx \frac{0.8451}{0.4771} \approx 1.771$

🎯 AMC-Style Worked Example

Problem: Solve $\log_2(x^2 - 4) - \log_2(x-2) = 3$ for $x$.

Solution:

  1. Domain check: $x^2 - 4 > 0$ and $x - 2 > 0$

    • $x^2 - 4 > 0$ → $x < -2$ or $x > 2$
    • $x - 2 > 0$ → $x > 2$
    • Combined: $x > 2$

  2. Use quotient law: $\log_2\left(\frac{x^2-4}{x-2}\right) = 3$

  3. Simplify fraction: $\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2$ (for $x \neq 2$)

  4. Convert: $x + 2 = 2^3 = 8$

  5. Solve: $x = 6$

  6. Check: $x = 6 > 2$ ✓ and $\log_2(36-4) - \log_2(6-2) = \log_2 32 - \log_2 4 = 5 - 2 = 3$ ✓

Answer: $x = 6$

⚠️ Common Pitfalls

Pitfall: Forgetting domain restrictions

Fix: Always check that all logarithm arguments are positive.

Pitfall: Wrong direction in inequalities

Fix: If base $0 < a < 1$, reverse the inequality direction.

Pitfall: Extraneous solutions

Fix: Check all solutions in the original equation, especially after squaring.

Pitfall: Confusing $\log_a(xy)$ and $\log_a x \cdot \log_a y$

Fix: $\log_a(xy) = \log_a x + \log_a y$ (addition, not multiplication).

📋 Quick Reference

Logarithm Laws

  • $\log_a(xy) = \log_a x + \log_a y$
  • $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
  • $\log_a(x^y) = y \log_a x$
  • $\log_a x = \frac{\log_b x}{\log_b a}$ (change of base)

Domain Restrictions

  • $\log_a x$ requires $x > 0$ and $a > 0, a \neq 1$
  • $\log_a(f(x))$ requires $f(x) > 0$

Inequality Rules

  • If $a > 1$: $\log_a x > \log_a y$ → $x > y$
  • If $0 < a < 1$: $\log_a x > \log_a y$ → $x < y$

Common Values

  • $\log 2 \approx 0.3010$
  • $\log 3 \approx 0.4771$
  • $\log 5 \approx 0.6990$
  • $\ln 2 \approx 0.693$

Next: Polynomial Roots & Vieta
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