Logarithmic graphs for modelling

A curved relationship can sometimes become linear after a logarithmic transformation. The two common forms need different horizontal axes:

Here log means base 10 unless stated otherwise. A graph of “Y against X” puts X horizontally and Y vertically. Always read the actual labels: the same plotted gradient does not mean the same model parameter on different axes.

For y = Axⁿ with A > 0 and x > 0:

log y = log A + n log x
Y = nX + c, where X = log x,
Y = log y and c = log A

The vertical intercept occurs at log x = 0, meaning x = 1. It does not represent x = 0. At that point y = A; whether the original formula is also defined at zero is a separate question.

For y = ABˣ, with A,B > 0:

log y = log A + x log B
Y = mx + c, where m = log B,
c = log A

A positive m gives B > 1 and growth; a negative m gives 0 < B < 1 and decay. If m = 0, B = 1 and the model is constant. Unlike the power graph, the horizontal variable is the original x, so zero can be included.

y = Ae^kx ⇒ ln y = ln A + kx
y = Axⁿ ⇒ ln y = ln A + n ln x

For ln(y) against x in the first model, the gradient is k and A = e^intercept. For ln(y) against ln(x) in the second, the gradient is n.

Using the same new base on both axes of an exact power relationship preserves its exponent. For an exponential model, the slope is ln B on a natural-log vertical axis and log₁₀B on a base-10 vertical axis. Do not use 10ᶜ to undo an ln intercept.

A plot of log y against log x passes through (0,1.2) and (2,4.2):

n = (4.2 − 1.2)/(2 − 0) = 1.5
log A = 1.2 ⇒ A = 10^1.2
y = 10^1.2x^1.5

At x = 4, this gives y = 8 · 10^1.2. The coordinates supplied by the graph were logarithms, not the original x and y values.

If y = Axⁿ with n ≠ 0, but the vertical axis is log x and the horizontal axis is log y, then:

log x = (1/n)log y − (log A)/n

The slope is now 1/n. If the displayed line is log x = m log y + c, then n = 1/m and log A = −c/m.

y = 4x³ ⇒ log x = ⅓log y − ⅓log 4

This reciprocal relationship describes the same exact line. Separately fitting noisy data with the two regression directions need not give reciprocal slopes, because the fitting errors are measured in different directions.

For cubes of edge length L, surface area S = 6L² and volume V = L³. Eliminating L gives S = 6V^2/3.

log S = log 6 + (2/3)log V
log V = (3/2)log S − (3/2)log 6

The first plot’s slope is 2/3; the reversed plot’s slope is 3/2. Neither changes the exponent in the original formula S = 6V^2/3.

You cannot take a real logarithm of zero or a negative measured value. Adding 1 to make an input positive changes the proposed model and needs justification; it is not a harmless plotting shortcut. Values between 0 and 1 are allowed and produce negative logarithms.

Use numerical values in stated, consistent units. Converting a length y from metres to centimetres multiplies it by 100 and adds 2 to log₁₀y. If x stays unchanged, a log–log slope is unchanged but the intercept and multiplier change.

For time models, specify the origin and unit. A constant k per day becomes k/24 per hour. On a power plot, log x = 0 means x = 1; on an exponential time plot, t = 0 is the chosen starting time.

Compare the transformed plots and the context. For exact evenly spaced x-values, a constant ratio in y suggests an exponential model; a power relationship instead has a constant proportional response to multiplying x.

Real data may only approximately follow either pattern. Check how the fitted model compares with the observations. Investigate unusual points rather than removing them simply to improve a line. Extrapolation needs a reason to expect the same mechanism and rate to continue.