🔢 Exponents & Logarithms

Exponents and logarithms are fundamental for AMC problems involving growth, decay, and equation solving. Master the laws and applications.

🎯 Key Ideas

Exponents: Represent repeated multiplication. $a^n = a \cdot a \cdot \ldots \cdot a$ ($n$ times)

Logarithms: Inverse operations of exponents. $\log_a x = y$ means $a^y = x$

Applications: Compound interest, population growth, radioactive decay, pH calculations.

⚡ Exponent Laws

Basic Laws

  • Product: $a^x \cdot a^y = a^{x+y}$
  • Quotient: $\frac{a^x}{a^y} = a^{x-y}$
  • Power: $(a^x)^y = a^{xy}$
  • Power of product: $(ab)^x = a^x b^x$
  • Power of quotient: $\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$

Special Cases

  • $a^0 = 1$ (for $a \neq 0$)
  • $a^{-x} = \frac{1}{a^x}$
  • $a^{1/x} = \sqrt[x]{a}$ (for $x > 0$)

Example: Simplify $\frac{2^{x+1} \cdot 8^{x-1}}{4^{2x}}$

Solution:

  1. Write with same base: $\frac{2^{x+1} \cdot (2^3)^{x-1}}{(2^2)^{2x}}$
  2. Apply power law: $\frac{2^{x+1} \cdot 2^{3(x-1)}}{2^{4x}}$
  3. Apply product law: $\frac{2^{x+1 + 3(x-1)}}{2^{4x}}$
  4. Simplify exponent: $\frac{2^{x+1 + 3x-3}}{2^{4x}} = \frac{2^{4x-2}}{2^{4x}}$
  5. Apply quotient law: $2^{(4x-2)-4x} = 2^{-2} = \frac{1}{4}$

📊 Logarithm Laws

Basic Laws

  • Product: $\log_a(xy) = \log_a x + \log_a y$
  • Quotient: $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
  • Power: $\log_a(x^y) = y \log_a x$
  • Change of base: $\log_a x = \frac{\log_b x}{\log_b a}$

Special Values

  • $\log_a 1 = 0$ (since $a^0 = 1$)
  • $\log_a a = 1$ (since $a^1 = a$)
  • $\log_a a^x = x$ (by definition)

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

Solution:

  1. Apply product law: $\log_2((x+1)(x-1)) = 3$
  2. Simplify: $\log_2(x^2-1) = 3$
  3. Convert to exponential: $x^2 - 1 = 2^3 = 8$
  4. Solve: $x^2 = 9$, so $x = \pm 3$
  5. Check domain: $x+1 > 0$ and $x-1 > 0$, so $x > 1$
  6. Solution: $x = 3$

🔄 Change of Base Formula

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

Common applications:

  • Convert to base 10: $\log_a x = \frac{\log x}{\log a}$
  • Convert to base $e$: $\log_a x = \frac{\ln x}{\ln a}$

Example: Evaluate $\log_3 7$ using base 10

Solution:

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

📈 Growth and Decay

Exponential Growth

  • Formula: $A(t) = A_0 e^{rt}$ or $A(t) = A_0 a^t$
  • Doubling time: $t = \frac{\ln 2}{r}$
  • Applications: Population growth, compound interest

Exponential Decay

  • Formula: $A(t) = A_0 e^{-rt}$ or $A(t) = A_0 a^t$ (where $0 < a < 1$)
  • Half-life: $t = \frac{\ln 2}{r}$
  • Applications: Radioactive decay, cooling

Example: A population grows at 3% per year. How long to double?

Solution:

  • Using $A(t) = A_0 e^{0.03t}$ and doubling time formula:
  • $t = \frac{\ln 2}{0.03} \approx \frac{0.693}{0.03} \approx 23.1$ years

🎯 AMC-Style Worked Example

Problem: Solve $2^{x+1} = 3^{x-1}$ for $x$.

Solution:

  1. Take logarithm of both sides: $\ln(2^{x+1}) = \ln(3^{x-1})$
  2. Apply power law: $(x+1)\ln 2 = (x-1)\ln 3$
  3. Expand: $x\ln 2 + \ln 2 = x\ln 3 - \ln 3$
  4. Collect $x$ terms: $x\ln 2 - x\ln 3 = -\ln 3 - \ln 2$
  5. Factor: $x(\ln 2 - \ln 3) = -(\ln 3 + \ln 2)$
  6. Solve: $x = \frac{-(\ln 3 + \ln 2)}{\ln 2 - \ln 3} = \frac{\ln 3 + \ln 2}{\ln 3 - \ln 2}$

Answer: $x = \frac{\ln 6}{\ln(3/2)}$

🔍 Common Traps & Fixes

Trap: Wrong logarithm domain

Fix: Remember $\log_a x$ requires $x > 0$ and $a > 0, a \neq 1$.

Trap: 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).

Trap: Forgetting to check solutions

Fix: Always verify solutions in original equation, especially for log equations.

Trap: Wrong base in change of base

Fix: $\log_a x = \frac{\log_b x}{\log_b a}$ (same base $b$ in numerator and denominator).

📋 Quick Reference

Essential Exponent Laws

  • $a^x \cdot a^y = a^{x+y}$
  • $\frac{a^x}{a^y} = a^{x-y}$
  • $(a^x)^y = a^{xy}$
  • $a^{-x} = \frac{1}{a^x}$

Essential Logarithm Laws

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

Common Values

  • $\log 2 \approx 0.3010$
  • $\log 3 \approx 0.4771$
  • $\log 5 \approx 0.6990$
  • $\ln 2 \approx 0.693$
  • $\ln 3 \approx 1.099$

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