AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.15 — Semi-log Plots
Notes — Linearizing Exponential Data
💡 Learning Objectives (2.15.A)
By the end of this lesson you will be able to:
- Construct a semi-log plot by graphing the logarithm of the output against the input
- Recognize that exponential data appear linear on a semi-log plot
- Read the slope and y-intercept of a semi-log plot to recover the original exponential model
- Use semi-log analysis to determine whether an exponential model is appropriate for given data
1. The Idea Behind a Semi-log Plot
Suppose y depends on x exponentially: y = a · b^x. Take the logarithm of both sides (any base; we'll use base 10 here):
log(y) = log(a) + x · log(b)
Let Y = log(y). The equation becomes Y = log(a) + log(b) · x — a LINEAR function of x with:
- slope = log(b)
- y-intercept = log(a)
So if we plot the original points (x, log(y)) — that is, leave x alone but logarithm the y-values — the result is a straight line whenever the data are exponential.
2. ‘Semi-log’ vs. ‘Log-log’
- Semi-log (or single-log) plot: only ONE axis is logarithmic, usually the y-axis. Useful for exponential data.
- Log-log plot: BOTH axes are logarithmic. Useful for power-function data y = a · x^k. (Outside the scope of this topic, but worth knowing.)
3. Example: Detecting an Exponential Pattern
Population data for a colony of bacteria, every hour:
t (hr) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
P (cells) | 200 | 500 | 1250 | 3125 | 7813 |
log(P) | 2.301 | 2.699 | 3.097 | 3.495 | 3.893 |
Each entry of P is 2.5 times the previous one, so log(P) grows by the same amount log(2.5) ≈ 0.398 per hour. The bottom row therefore lies on a straight line.
4. Reading a Model off a Semi-log Line
📘 Example — Recover y = a · b^x from semi-log fit Y = 1.4 + 0.30 x
- Y = log(y), so log(y) = 1.4 + 0.30 x
- Equate to log(a) + x · log(b): log(a) = 1.4 and log(b) = 0.30
- Solve: a = 10^1.4 ≈ 25.12 and b = 10^0.30 ≈ 1.995
- Original model: y ≈ 25.12 · (1.995)^x
5. Reasoning About the Slope
Because slope = log(b), the SIGN of the slope on a semi-log plot tells you whether the underlying quantity grows or decays:
- Positive slope ⇒ log(b) > 0 ⇒ b > 1 ⇒ exponential growth
- Zero slope ⇒ log(b) = 0 ⇒ b = 1 ⇒ constant
- Negative slope ⇒ log(b) < 0 ⇒ b < 1 ⇒ exponential decay
The MAGNITUDE of the slope is directly related to the growth rate: a steeper line means a more rapidly changing exponential.
6. Why Semi-log Plots Are Useful in Practice
- Compress huge ranges: data spanning many orders of magnitude (microbes, finance, radioactive isotopes) fit on one screen.
- Quick visual diagnostic: if a curved scatter looks linear after taking logs of y, exponential is a reasonable model.
- Easy parameter recovery: linear fits are simple and well-understood, while fitting an exponential directly is harder.
7. A Cautionary Note
⚠️ Common mistake
Just because a few points look ‘sort of linear’ on a semi-log plot does not guarantee an exponential model. Look at residuals or test additional points before committing.
Also remember: log of a non-positive number is undefined. If your data contain zero or negative y-values, you cannot put them on a semi-log axis.
8. Summary
- A semi-log plot graphs log(y) against x
- Exponential data y = a · b^x become a straight line with slope log(b) and intercept log(a)
- Read the parameters off the line and back-transform to recover a and b
- Slope sign tells growth (positive), constancy (zero), or decay (negative)
- Useful for compressing wide ranges and diagnosing whether an exponential fits