Logarithmic representation

Upper scale divided into linear units Lower scale divided into logarithmic units${\ displaystyle \ lg x}$
${\ displaystyle x}$
Distribution of the number of articles by the most active authors in the German-language Wikipedia

The logarithmic representation uses an axis labeling in which the numerical value of a quantity to be represented is not plotted in a linear division, but the logarithm of its numerical value. In a chart , this representation is applied to one or both axes.

Such a representation is particularly helpful when the range of values ​​of the data represented covers many orders of magnitude . The logarithmic representation makes relationships in the area of ​​small values ​​easier to understand. Various mathematical relationships can be better illustrated or made recognizable through a logarithmic representation.

In principle, the same distances reflect the same factors in the direction of the logarithmic axis; If one distance corresponds to the factor 10, then twice the distance in the diagram corresponds to the factor 10 2 .

The same distribution in a simple logarithmic representation
The same distribution in double logarithmic representation

Usual display options

Bode diagram of a low-pass filter:
above phase frequency response single logarithmic,
below amplitude frequency response double logarithmic
Growth of the population of England on a logarithmic scale (1.67 decade).

If numerical relationships are in the foreground, the decadic logarithm is used; the natural logarithm is used for a more fundamental view .

The illustrated Bode diagram shows the transfer function of a low-pass filter over a frequency range of more than four powers of ten as an application in electrical engineering .

Especially before the introduction of computer graphics, logarithm paper was an important aid for representation. Single-logarithmic paper or double-logarithmic paper are available for drawing diagrams in logarithmic representation . The possibilities of graphic representations on the computer have simplified the use of logarithmic scales and greatly reduced the use of such paper.

Mathematical modeling

With the transition to a new variable

${\ displaystyle X = \ log (x)}$ and ${\ displaystyle Y = \ log (y)}$

For some functions , the presentation is simplified and certain relationships are illustrated. Conversely, a straight course in a sequence of measuring points with suitably divided axes allows conclusions to be drawn about the underlying function.

A power law becomes a straight line in a double logarithmic representation

${\ displaystyle y = a \, x ^ {b} \, \ Rightarrow \; Y = \ log (a) + bX}$

The special case of a straight line falling to the right at an angle of 45 ° (with the same scales on both axes) shows , that is, inverse proportionality . ${\ displaystyle b = -1}$

An exponential curve can be represented in a simple logarithmic representation as a straight line

${\ displaystyle y = a \, \ mathrm {e} ^ {bx} \, \ Rightarrow \; Y = \ log (a) + bx \ cdot \ log (\ mathrm {e})}$

A function in the form of a normal distribution (Gaussian bell curve) becomes a parabola in a simple logarithmic representation

${\ displaystyle y = a \, \ mathrm {e} ^ {bx ^ {2}} \, \ Rightarrow \; Y = \ log (a) + bx ^ {2} \ cdot \ log (\ mathrm {e} )}$

A function of the form of a lognormal distribution is in simple logarithmic representation to a normal distribution and in double logarithmic representation to a parable

${\ displaystyle y = a \, \ mathrm {e} ^ {b (\ log (x)) ^ {2}} \, \ Rightarrow \; y = a \, \ mathrm {e} ^ {bX ^ {2 }}}$
${\ displaystyle y = a \, \ mathrm {e} ^ {b (\ log (x)) ^ {2}} \, \ Rightarrow \; Y = \ log (a) + bX ^ {2} \ cdot \ log (\ mathrm {e})}$

Other uses

For certain tasks it has become common practice to scale an axis as the logarithm of the logarithm (log (log (y))), for example the vertical axis in the graphic representation of oil viscosities according to Ubbelohde-Walther or in the Weibull probability paper. Here the double-logarithmic axis is sometimes referred to as double-logarithmic.

Individual evidence

1. Quality management in the automotive industry - 3, "Assurance of Reliability for Automobile Manufacturers and Suppliers", published by VDA 2000, , Section 2.4.3