Logarithmic scale ( $u = lo g_{b} (x / x_{0})$ )

still means the original value one hundred. Ordinary real logarithms exclude zero and negative inputs. Omitting such values or clipping them to a plotting boundary does not give them genuine logarithmic positions; the plotting choice needs to be clear.^[ref]

Table of contents

Order of magnitude ( $Δ = lo g_{10} (x_{2} / x_{1})$ )

 1  0

For positive quantities, a difference of one order of magnitude means a factor of ten; a difference of

n

orders of magnitude means a factor of

1 0^{n}

. The logarithmic difference

lo g_{10} (x_{2} / x_{1})

records this comparison, without subtracting the quantities themselves. For example,

1000

is three orders of magnitude above one, regardless of a common change of units. Roughly assigning a single value to an order of magnitude also requires a rounding convention; powers of ten are the reference levels.

Logarithmic midpoint ( $m = ab$ )

 0  0

For positive endpoints

a, b

, halfway along a logarithmic scale means

lo g_{β} (m / x_{0}) = \frac{l o g _{β} ( a / x _{0} ) + l o g _{β} ( b / x _{0} )}{2} .

(1)

Here

β > 1

is the base. Equivalently, using the natural logarithm,

2 ln (m / x_{0}) = ln (a / x_{0}) + ln (b / x_{0})

. The product of numbers identity for logarithms gives

m^{2} = ab

, so the midpoint is the geometric mean $ab$ . The two endpoints lie at equal multiplicative distances because

m / a = b / m

For

a = 10

and

b = 100

, this midpoint is

1000 \approx 31.6228

, giving a factor of

10

on each side. The arithmetic mean

55

is halfway on a linear scale, but its two ratios are

55/10 = 5.5

and

100/55 \approx 1.81818

. Thus reading halfway between the marks

10

and

100

55

on a logarithmic scale gives the wrong value.

Figure 1.
Equal ratios give equal logarithmic gaps, and the geometric mean is halfway between two positive values
. Original rulers calculated from logarithms. The lower rulers cover the same endpoints $10$ and $100$ : the blue geometric mean is centered on the logarithmic scale, while the orange arithmetic mean is centered on the linear scale. Tick labels show original values. **Details**.

Semi-log plot

 1  0

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]

Log-log plot

 1  0

A log-log plot uses logarithmic scales on both axes. For a positive power law

y = C x^{p}

with

x, C > 0

, positive references

x_{0}, y_{0}

give

lo g_{b} (y / y_{0}) = lo g_{b} (C x_{0}^{p} / y_{0}) + p lo g_{b} (x / x_{0}) .

(1)

Its transformed slope is the exponent

p

when both axes use the same logarithm base and the slope is measured in logarithmic coordinate units. For example,

y = 2 x^{2}

takes the values

2, 8, 32

x = 1, 2, 4

. Every doubling of

x

quadruples

y

, giving slope two on a log-log plot.

This differs from a semi-log plot: a power law has constant proportional response to a proportional change of input, whereas exponential growth has constant proportional response to an additive change of input. Both can appear straight with appropriate axes, so one must read the scale labels before interpreting a slope.

 View more

A logarithmic scale is a way of displaying numerical data over a wide range of values in a way that can make it easier to visualize and interpret. Instead of each unit being the same size as on a linear scale, where equal intervals on the axis represent equal differences in value, a logarithmic scale represents equal intervals as equal ratios. In a logarithmic scale, each tick mark on the axis represents a power of a base number, commonly 10.

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Logarithmic scale ( $u = lo g_{b} (x / x_{0})$ )

Order of magnitude ( $Δ = lo g_{10} (x_{2} / x_{1})$ )

Logarithmic midpoint ( $m = ab$ )

Semi-log plot

Log-log plot

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