🔢 Exponential & Log Tricks — Growth Comparison & Base Alignment
Essential techniques for solving exponential and logarithmic problems efficiently.
🎯 Recognition Cues
- “Solve for $x$” — Exponential or logarithmic equations
- “Compare” — Growth rate comparisons
- “Find the value” — Complex exponential/logarithmic expressions
- $a^x = b$ — Exponential equations
- $\log_a x = b$ — Logarithmic equations
📚 Template Solutions
Exponential Equations
| Type | Method | Example |
|---|---|---|
| Same base | $a^x = a^y$ → $x = y$ | $2^x = 2^3$ → $x = 3$ |
| Different base | Take logarithm | $2^x = 3$ → $x = \frac{\log 3}{\log 2}$ |
| Quadratic form | Substitute $u = a^x$ | $2^{2x} - 5 \cdot 2^x + 6 = 0$ → $u^2 - 5u + 6 = 0$ |
Logarithmic Equations
| Type | Method | Example |
|---|---|---|
| Same base | $\log_a x = \log_a y$ → $x = y$ | $\log_2 x = \log_2 3$ → $x = 3$ |
| Different base | Use change of base | $\log_2 x = \log_3 9$ → $x = 2^{\log_3 9}$ |
| Product/quotient | Use log properties | $\log(xy) = \log x + \log y$ |
Growth Comparison
| Technique | When to Use | Example |
|---|---|---|
| Compare bases | Same exponent | $2^3$ vs $3^3$ |
| Compare exponents | Same base | $2^5$ vs $2^3$ |
| Use logarithms | Different base and exponent | $2^3$ vs $3^2$ |
🎯 Worked Examples
Example 1: Basic Exponential Equation
Problem: Solve $2^x = 8$
Solution:
- Method 1: Express as same base: $2^x = 2^3$ → $x = 3$
- Method 2: Take logarithm: $x = \frac{\log 8}{\log 2} = \frac{\log 2^3}{\log 2} = \frac{3\log 2}{\log 2} = 3$
- Answer: $x = 3$
Example 2: Quadratic Form
Problem: Solve $3^{2x} - 4 \cdot 3^x + 3 = 0$
Solution:
- 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: Growth Comparison
Problem: Compare $2^{100}$ and $100^2$
Solution:
- Take logarithms: $\log(2^{100}) = 100\log 2 \approx 100 \cdot 0.301 = 30.1$
- Take logarithms: $\log(100^2) = 2\log 100 = 2 \cdot 2 = 4$
- Compare: $30.1 > 4$, so $2^{100} > 100^2$
- Answer: $2^{100} > 100^2$
⚠️ Common Pitfalls
Pitfall: Forgetting domain restrictions
- Fix: Check that arguments of logarithms are positive
- Example: $\log(x-1)$ requires $x > 1$
Pitfall: 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 $\log 2 + \log 3 = \log 6$
Pitfall: 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:
- Factor out common term: $2^{x-1}(2^2 + 1) = 5$
- Simplify: $2^{x-1} \cdot 5 = 5$
- Divide by 5: $2^{x-1} = 1$
- Express as same base: $2^{x-1} = 2^0$
- Equate exponents: $x-1 = 0$
- Solve: $x = 1$
- Verify: $2^{1+1} + 2^{1-1} = 2^2 + 2^0 = 4 + 1 = 5$ ✓
- Answer: $x = 1$
Key insight: Factoring out common exponential terms often simplifies the equation.
🔗 Related Patterns
- 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
Next: Word Problems | Prev: Functional Equation Templates | Back: Problem Types Overview