Semi-log plot

A semi-log plot is a type of graph used to visualize data that spans several orders of magnitude. In a semi-logarithmic plot, one axis (usually the y-axis) is scaled logarithmically, while the other axis (usually the x-axis) is scaled linearly. This method of plotting is particularly useful for displaying data that follows exponential growth or decay patterns, as it can make trends easier to identify and interpret.

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A semi-log plot uses a logarithmic scale on one axis and a linear scale on the other. Which axis is logarithmic matters. Plotting

x

against

lo g y

makes positive exponential functions appear as straight lines; plotting

lo g x

against

y

instead makes an affine function of a logarithm appear straight. The plotting operations change positions while retaining the original tick values.^[ref]

Take the original dimensionless mathematical model

y (t) = 3 e^{0.4 t}

for

0 \leq t \leq 8

. It has exponential growth, with

y (0) = 3

y (4) \approx 14.8591

, and

y (8) \approx 73.5976

. Taking a base-ten logarithm gives

lo g_{10} y = lo g_{10} 3 + \frac{0.4}{l n 10} t .

(1)

The slope in these transformed positions is

0.4/ ln 10 \approx 0.173718

, although the original relative growth rate

y^{'} / y

0.4

. Equal steps of two in

t

multiply

y

e^{0.8} \approx 2.22554

. The corresponding ordinary gains increase, so the left plot bends while the right plot is straight.

Figure 1.
The same exponential growth model bends on a linear scale and is straight on a semi-log plot with a logarithmic vertical axis
. Original exact mathematical model $y (t) = 3 e^{0.4 t}$ , not measurements. Both panels show identical values at the same $t$ . Right-hand tick labels retain the original $y$ values, with equal gaps for each doubling. The stated slope is for $(t, lo g_{10} y)$ , not the original derivative $y^{'}$ . **Details**.

One practical use is checking a proposed exponential distribution for failure times. For rate

λ > 0

and

t \geq 0

, its cumulative distribution function is

F (t) = 1 - e^{- λ t}

, so the reciprocal survival function

y = 1/ [1 - F (t)]

satisfies

lo g_{10} y = λ t / ln 10

. Estimated probabilities can therefore be placed on a logarithmic vertical axis and compared with a straight line. Agreement supports compatibility with the chosen statistical model; a roughly straight finite sample does not establish a unique physical mechanism.^[ref]

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Semi-log plot

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