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Semi-Log Plots: Linearizing Exponential Data

You’ll be able to

Take a log and the curve becomes a line

Start with y = a·bˣ and take the log of both sides: log y = log a + x·log b. Read that as a linear equation in x whose output is log y. Its slope is log b and its vertical intercept is log a. So plotting log y against x turns exponential data into a straight line — and straightness is far easier to judge by eye than exponentiality. This is a semi-log plot: one axis logarithmic, one linear.

Linearizing an exponential
y = a·bˣ ⟹ log y = (log b)·x + log a
Slope = log b, so b = 10^slope. Intercept = log a, so a = 10^intercept. Using ln instead, b = e^slope and a = e^intercept.
Worked example

A semi-log plot of log₁₀(y) against x is linear with slope 0.30 and intercept 1.70. Find the exponential model.

  1. 1.log y = 0.30x + 1.70.
  2. 2.Undo the logarithm: y = 10^(0.30x + 1.70) = 10^1.70 · 10^(0.30x).
  3. 3.10^1.70 ≈ 50.1, so a ≈ 50.
  4. 4.10^0.30 ≈ 1.995, so b ≈ 2.
  5. 5.Model: y ≈ 50·2ˣ.
Answer: y ≈ 50·2ˣ. The slope of 0.30 is recognizable — log₁₀ 2 ≈ 0.301 — so a semi-log slope near 0.30 is the fingerprint of doubling, and one near 1 means a factor of ten per step.

Why this is worth doing

Two reasons, both practical. First, diagnosis: whether a set of points lies on a line is something the human eye judges reliably, whereas "is this curve exponential or is it a high-degree polynomial?" is not. Second, range: data spanning many orders of magnitude — bacterial counts from 10 to 10⁸ — is unreadable on a linear axis, where everything below the maximum is squashed against zero. A log axis gives every decade the same width.

Tip

On a semi-log plot, equal vertical distances mean equal ratios, not equal differences. The gap from 1 to 10 is the same size as the gap from 10 to 100 and from 1,000 to 10,000. Reading such an axis as if it were linear misjudges values by orders of magnitude.

Checkpoint

Data plotted with log(y) against x falls on a straight line with negative slope. What kind of relationship is this?

Semi-log versus log-log

Which plot straightens the data tells you which family you have — a genuinely useful diagnostic. Semi-log (log y vs x) straightens exponentials, y = a·bˣ, because taking a log makes the exponent linear. Log-log (log y vs log x) straightens power laws, y = a·xⁿ, since log y = log a + n·log x, and the slope there is the exponent n. So: if semi-log is straight, the growth is exponential; if log-log is straight, it is a power law.

Which plot straightens which family
exponential y = a·bˣ → semi-log is linear · power law y = a·xⁿ → log-log is linear, with slope n
A quick discriminator: exponentials eventually beat every power law, and only the exponential has a constant ratio over equal x-steps.
Worked example

Measured y at x = 1, 2, 3, 4 is 6.0, 24.0, 54.0, 96.0. Is this exponential or a power law?

  1. 1.Ratios of successive y: 24/6 = 4, 54/24 = 2.25, 96/54 ≈ 1.78. Not constant, so not exponential.
  2. 2.Try a power law: check y/x². 6/1 = 6, 24/4 = 6, 54/9 = 6, 96/16 = 6. Constant.
  3. 3.So y = 6x², a power law with n = 2.
  4. 4.A log-log plot would be a line of slope 2; a semi-log plot would be visibly curved.
Answer: It is a power law, y = 6x². The ratio test rules out exponential immediately — a genuine exponential would multiply by the same factor at each equal step in x, and 4, 2.25, 1.78 does not.
Checkpoint

A log-log plot of y against x is a straight line with slope 3. What is the relationship?

Answer the 2 checkpoints as you read.

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