🔢 Exponential & Log Equations — Change of Base & Growth

Essential for AMC12 problems involving exponential and logarithmic functions.

🎯 Key Ideas

Exponential Equations — Equations with variables in exponents, solved using logarithms or by expressing both sides with the same base.

Logarithmic Equations — Equations with logarithms, solved using logarithm properties and change of base.

Change of Base — $\log_a x = \frac{\log_b x}{\log_b a}$ allows conversion between different logarithm bases.

📊 Essential Concepts

Exponential Equations

TypeMethodExample
Same base$a^x = a^y$ → $x = y$$2^x = 2^3$ → $x = 3$
Different baseTake logarithm of both sides$2^x = 3$ → $x = \frac{\log 3}{\log 2}$
Quadratic formSubstitute $u = a^x$$2^{2x} - 5 \cdot 2^x + 6 = 0$ → $u^2 - 5u + 6 = 0$

Logarithmic Equations

TypeMethodExample
Same base$\log_a x = \log_a y$ → $x = y$$\log_2 x = \log_2 3$ → $x = 3$
Different baseUse change of base$\log_2 x = \log_3 9$ → $x = 2^{\log_3 9}$
Product/quotientUse log properties$\log(xy) = \log x + \log y$

Change of Base

FormulaExample
$\log_a x = \frac{\log_b x}{\log_b a}$$\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} = \frac{0.903}{0.301} = 3$
$\log_a x = \frac{\ln x}{\ln a}$$\log_3 27 = \frac{\ln 27}{\ln 3} = \frac{3.296}{1.099} = 3$

🎯 Micro-Examples

Example 1: Solve $2^x = 8$

  • Method 1: Express as same base: $2^x = 2^3$ → $x = 3$
  • Method 2: Take logarithm: $x = \frac{\log 8}{\log 2} = \frac{\log 2^3}{\log 2} = \frac{3\log 2}{\log 2} = 3$

Example 2: Solve $3^{2x} - 4 \cdot 3^x + 3 = 0$

  • Substitute $u = 3^x$: $u^2 - 4u + 3 = 0$
  • Factor: $(u-1)(u-3) = 0$ → $u = 1$ or $u = 3$
  • Solve: $3^x = 1$ → $x = 0$; $3^x = 3$ → $x = 1$
  • Answer: $x = 0$ or $x = 1$

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

  • Domain: $x+1 > 0$ and $x-1 > 0$ → $x > 1$
  • Use log property: $\log_2[(x+1)(x-1)] = 3$
  • Simplify: $\log_2(x^2-1) = 3$
  • Exponentiate: $x^2-1 = 2^3 = 8$
  • Solve: $x^2 = 9$ → $x = \pm 3$
  • Check domain: $x = 3 > 1$ ✓; $x = -3 < 1$ ✗
  • Answer: $x = 3$

⚠️ Common Traps & Fixes

Trap: Forgetting domain restrictions

  • Fix: Check that arguments of logarithms are positive
  • Example: $\log(x-1)$ requires $x > 1$

Trap: Incorrect logarithm properties

  • Fix: Remember $\log(xy) = \log x + \log y$, not $\log(x+y) = \log x + \log y$
  • Example: $\log(2+3) = \log 5$, not $\log 2 + \log 3 = \log 6$

Trap: Sign errors in change of base

  • Fix: $\log_a x = \frac{\log_b x}{\log_b a}$, not $\frac{\log_b a}{\log_b x}$
  • Example: $\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2}$, not $\frac{\log_{10} 2}{\log_{10} 8}$

🎯 AMC-Style Worked Example

Problem: Solve $2^{x+1} + 2^{x-1} = 5$.

Solution:

  1. Factor out common term: $2^{x-1}(2^2 + 1) = 5$
  2. Simplify: $2^{x-1} \cdot 5 = 5$
  3. Divide by 5: $2^{x-1} = 1$
  4. Express as same base: $2^{x-1} = 2^0$
  5. Equate exponents: $x-1 = 0$
  6. Solve: $x = 1$
  7. Verify: $2^{1+1} + 2^{1-1} = 2^2 + 2^0 = 4 + 1 = 5$ ✓
  8. Answer: $x = 1$

Key insight: Factoring out common exponential terms often simplifies the equation.

🔗 Related Topics

  • Exponent Rules — Essential for manipulating exponential expressions
  • Logarithm Properties — Needed for solving logarithmic equations
  • Domain — Exponential and logarithmic functions have domain restrictions
  • Growth — Exponential functions model growth and decay

📝 Practice Checklist

  • Master exponential equation solving
  • Practice logarithmic equation solving
  • Learn change of base technique
  • Practice domain restrictions
  • Understand growth comparisons
  • Work on speed and accuracy

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