AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.11 — Logarithmic Functions
Notes — Graphs and Behavior
💡 Learning Objectives (2.11.A)
By the end of this lesson you will be able to:
- Identify the domain, range, and end behavior of a logarithmic function
- Locate the vertical asymptote and key reference point of f(x) = log_b(x)
- Describe whether the function is increasing or decreasing based on the base
- Analyze the rate of change and concavity of a logarithmic function
1. The Parent Logarithmic Function
The parent logarithmic function is f(x) = log_b(x) where b > 0 and b ≠ 1. Two cases produce different shapes — one increasing, one decreasing — depending on whether the base is greater or less than 1.
2. Domain, Range, and Asymptote
- Domain: x > 0 (the argument must be positive)
- Range: all real numbers
- Vertical asymptote: x = 0 (the y-axis)
- There is no horizontal asymptote
- Every logarithmic function f(x) = log_b(x) passes through (1, 0) and (b, 1)
📝 Why x = 0 is the asymptote
As x approaches 0 from the right, you are asking ‘what exponent on b gives a number very close to 0?’ For b > 1 the answer is a very negative exponent; for 0 < b < 1 the answer is a very positive exponent. Either way the output blows up in magnitude.
3. Increasing or Decreasing — It Depends on the Base
- If b > 1, then f(x) = log_b(x) is increasing on (0, ∞). Outputs grow toward +∞ as x grows.
- If 0 < b < 1, then f(x) = log_b(x) is decreasing on (0, ∞). Outputs grow toward −∞ as x grows.
This mirrors what happens with the corresponding exponential function: an exponential with base > 1 grows, and so does its inverse log; an exponential with base between 0 and 1 decays, and so does its inverse log.
4. End Behavior
For b > 1:
- As x → 0⁺, f(x) → −∞
- As x → ∞, f(x) → ∞ (but very slowly)
For 0 < b < 1:
- As x → 0⁺, f(x) → ∞
- As x → ∞, f(x) → −∞
5. Rate of Change and Concavity
Even when a logarithmic function is increasing, the rate at which it increases gets smaller and smaller. Doubling x adds the same fixed amount to the output (specifically, log_b(2)), regardless of how big x already is. So a one-unit change in y happens over wider and wider intervals of x.
- For b > 1: f(x) = log_b(x) is increasing AND concave down on (0, ∞). Rate of change is positive but decreasing.
- For 0 < b < 1: f(x) = log_b(x) is decreasing AND concave up on (0, ∞). Rate of change is negative but increasing toward 0.
⚠️ Common mistake
Concave down does NOT mean decreasing. The natural log y = ln(x) is increasing on (0, ∞) yet still concave down — the curve smiles less and less as it grows.
6. A Comparison Table
x | 0.5 | 1 | 2 | 4 | 8 | 16 |
|---|---|---|---|---|---|---|
log₂(x) | −1 | 0 | 1 | 2 | 3 | 4 |
log₁/₂(x) | 1 | 0 | −1 | −2 | −3 | −4 |
log₂(x) climbs slowly to the right and dives toward −∞ near x = 0. log₁/₂(x) is its mirror image — declining slowly to the right, climbing to +∞ near 0.
7. Summary
- Domain x > 0; range all reals; vertical asymptote x = 0
- Always passes through (1, 0) and (b, 1)
- Increasing for b > 1; decreasing for 0 < b < 1
- Always changes concavity in the same direction across its whole domain
- Logarithmic growth is slower than any positive power of x