Module 4: Quick Reference Card
Exponential and Logarithmic Functions - One-Page Cheat Sheet
Exponential Functions
General Form
f(x) = a·bx
- a = initial value
- b = base (growth/decay factor)
- b > 1: exponential growth
- 0 < b < 1: exponential decay
Key Properties
- Domain: (-∞, ∞)
- Range: (0, ∞)
- Y-intercept: (0, a)
- Horizontal asymptote: y = 0
Natural Exponential
f(x) = ex
e ≈ 2.71828
Base for continuous growth/decay
Compound Interest
Periodic: A = P(1 + r/n)nt
Continuous: A = Pert
P = principal, r = rate, n = compounds/year, t = time
Logarithmic Functions
Definition
y = log_b(x) ⟺ by = x
Logarithm = inverse of exponential
Common log: log(x) = log₁₀(x)
Natural log: ln(x) = log_e(x)
Basic Properties
- log_b(1) = 0
- log_b(b) = 1
- log_b(bx) = x
- blog_b(x) = x
- ln(1) = 0
- ln(e) = 1
Graph Properties
- Domain: (0, ∞)
- Range: (-∞, ∞)
- X-intercept: (1, 0)
- Vertical asymptote: x = 0
Logarithm Properties
Product Rule: log_b(MN) = log_b(M) + log_b(N)
Quotient Rule: log_b(M/N) = log_b(M) - log_b(N)
Power Rule: log_b(Mp) = p·log_b(M)
Change of Base: log_b(x) = ln(x)/ln(b)
Solving Equations
Exponential Equations
Same base:
If bm = bn, then m = n
Different bases:
- Isolate exponential
- Take ln of both sides
- Use power rule
- Solve for variable
Logarithmic Equations
- Combine logs using properties
- Convert to exponential form
- Solve equation
- Check: arguments must be > 0
Always check for extraneous solutions!
Application Formulas
Growth and Decay
Growth: P(t) = P₀ert
Decay: A(t) = A₀e-kt
Half-life: t1/2 = ln(2)/k
ln(2) ≈ 0.693
Other Applications
pH: pH = -log[H+]
[H+] = 10-pH
Cooling:
T(t) = Troom + (T₀ - Troom)e-kt
Common Values to Know
Exponential Values
| 23 = 8 | 24 = 16 | 25 = 32 |
| 32 = 9 | 33 = 27 | 34 = 81 |
| 42 = 16 | 52 = 25 | 53 = 125 |
Logarithm Values
| log₂(8) = 3 | log₃(9) = 2 |
| log₂(16) = 4 | log₃(27) = 3 |
| log(10) = 1 | log(100) = 2 |
| log(1000) = 3 | ln(e) = 1 |
Important Constants
| e ≈ 2.71828 | ln(2) ≈ 0.6931 | ln(10) ≈ 2.3026 |
Graph Characteristics
y = bx (b > 1)
- Passes through (0, 1)
- Increases from left to right
- Approaches y = 0 as x → -∞
- Grows rapidly as x → ∞
y = log_b(x) (b > 1)
- Passes through (1, 0)
- Increases from left to right
- Approaches x = 0 from right
- Grows slowly as x → ∞
Transformation Quick Guide
| Transformation | Form | Effect |
|---|---|---|
| Vertical shift | f(x) + k | Up k units (down if k < 0); asymptote shifts |
| Horizontal shift | f(x - h) | Right h units (left if h < 0) |
| Vertical reflection | -f(x) | Flip over x-axis |
| Horizontal reflection | f(-x) | Flip over y-axis |
Common Mistakes to Avoid
- WRONG: log_b(M + N) = log_b(M) + log_b(N) - There is NO product rule for sums!
- WRONG: log_b(M - N) = log_b(M) - log_b(N) - Quotient rule only for division!
- WRONG: (log M)(log N) = log(MN) - Don't confuse multiplication with product rule!
- Remember: Always check that logarithm arguments are positive!
- Remember: The base of an exponential function must be positive and ≠ 1!
Problem-Solving Strategies
Expanding Logarithms
- Identify products → use + rule
- Identify quotients → use - rule
- Identify powers → bring down exponent
- Simplify special logs (log_b(b) = 1)
Condensing Logarithms
- Move coefficients as exponents
- Combine + terms as products
- Combine - terms as quotients
- Write as single logarithm