Overview
Exponential functions model growth and decay processes and appear frequently in SAT Advanced Math, especially in word problems involving populations, investments, and radioactive decay. The SAT tests interpreting exponential models, evaluating functions for given inputs, and solving exponential equations. Logarithms appear as a tool to solve exponential equations where bases cannot be easily matched. The digital SAT’s Desmos tool makes evaluating and graphing exponential functions efficient.
Key Points
1. Standard Exponential Function Form
f(x) = a · bˣ
| Parameter | Meaning |
|---|---|
| a | Initial value (y-intercept); f(0) = a |
| b | Growth/decay factor (base) |
| b > 1 | Exponential growth |
| 0 < b < 1 | Exponential decay |
| x | Time or input variable |
Converting percent to base:
- Growth rate of p%: b = 1 + p/100
- Decay rate of p%: b = 1 – p/100
- Example: 15% decay → b = 0.85 (not 0.15)
2. Compound Interest Formula
A = P(1 + r/n)^(nt)
| Variable | Meaning |
|---|---|
| A | Final amount |
| P | Principal (initial amount) |
| r | Annual interest rate (decimal) |
| n | Number of compoundings per year |
| t | Time in years |
Special case (continuous compounding): A = Pe
3. Properties of Exponents
a^m · a^n = a^(m+n) ← Product rule
a^m / a^n = a^(m-n) ← Quotient rule
(a^m)^n = a^(mn) ← Power rule
a^0 = 1 ← Zero exponent
a^(-n) = 1/a^n ← Negative exponent
(ab)^n = a^n · b^n ← Product to power
(a/b)^n = a^n / b^n ← Quotient to power
4. Solving Exponential Equations
Method 1: Make bases equal (when possible)
3^x = 81 → 3^x = 3^4 → x = 4
Method 2: Apply logarithm to both sides
2^x = 10 → x·log(2) = log(10) → x = log(10)/log(2) ≈ 3.32
Method 3: Recognize the structure (SAT common pattern)
f(x) = 23(1.20)^(x/4)
Increase x by 8: new exponent increases by 8/4 = 2
Growth factor for that change: (1.20)² = 1.44 → 44% increase
5. Logarithm Properties
log_b(xy) = log_b(x) + log_b(y) ← Product rule
log_b(x/y) = log_b(x) - log_b(y) ← Quotient rule
log_b(x^n) = n · log_b(x) ← Power rule
log_b(b) = 1 ← Base rule
log_b(1) = 0 ← Zero rule
log_b(x) = log(x) / log(b) ← Change of base
Logarithm is the inverse of exponentiation:
b^y = x ↔ log_b(x) = y
6. Linear vs. Exponential Growth
| Feature | Linear | Exponential |
|---|---|---|
| Rate of change | Constant (adds) | Multiplicative (multiplies) |
| Equation form | f(x) = mx + b | f(x) = a·bˣ |
| Table pattern | Constant differences | Constant ratios |
| Graph shape | Straight line | Curved (increasing or decreasing) |
Pitfalls and Common Mistakes
Pitfall 1: Converting percent incorrectly “A population decays at 15% per year” — students write b = 0.15 or b = 1.15. Fix: A 15% decay means 85% remains each year → b = 0.85. Growth of 15% → b = 1.15.
Pitfall 2: Confusing power functions with exponential functions x² (power function, variable is the base) vs. 2^x (exponential function, variable is the exponent). Their behaviors are very different for large x. Fix: Look at where the variable appears. Variable in exponent → exponential.
Pitfall 3: Doubling-time misconception “After 2 time periods, the amount doubles twice.” Wrong — it quadruples (multiplies by b² if b=2). Fix: Use the formula, not intuition. f(2t) = a·b^(2t) = a·(b²)
Pitfall 4: Incorrect initial value Students use f(1) = ab as the initial value instead of f(0) = a. Fix: The y-intercept and initial value are f(0) = a·b^0 = a·1 = a.
Pitfall 5: Linear thinking in exponential context In a table where values double each row, students add a constant difference instead of recognizing the multiplicative (exponential) pattern. Fix: Check if the ratio of consecutive outputs is constant. If yes → exponential.
Related Entries
- Radicals_Rational_Exponents
- Functions_Concepts_Notation
- Function_Transformations
- Quadratic_Functions_Parabolas
- Polynomial_Operations_Factoring
Quick Reference Card
| Formula / Rule | Expression |
|---|---|
| Standard form | f(x) = a·bˣ |
| Initial value | a = f(0) |
| Growth factor | b = 1 + r (r as decimal) |
| Decay factor | b = 1 - r (r as decimal) |
| Compound interest | A = P(1+r/n)^(nt) |
| Product rule | a^m · a^n = a^(m+n) |
| Power rule | (a^m)^n = a^(mn) |
| Negative exponent | a^(-n) = 1/aⁿ |
| Log inverse | b^y = x ↔ log_b(x) = y |
| Change of base | log_b(x) = log(x)/log(b) |
| Linear indicator | constant differences in table |
| Exponential indicator | constant ratios in table |