Logarithmic scale
The logarithmic representation is based on a scale that does not use the numerical value of a physical quantity, but the logarithm of their number value. In logarithmic representation of a graph showing the values of one or more axes are plotted the logarithm. Such a representation is particularly useful when the range of the data shown includes many orders of magnitude. Due to the logarithmic representation of relationships in the area of small values are better manageable.
In principle, the same distances play in the direction of the logarithmic axis same factors; therefore corresponds to a distance by a factor of 10, then corresponds to twice the distance in the graph by a factor of 102
Usual display options
If numerical relationships are at the forefront, working with the logarithm; with more fundamental consideration of the natural logarithm is used.
The illustrated Bode plot shows an application in the electrical transfer function of a low-pass filter over a frequency range of more than four orders of magnitude.
Especially before the advent of computer graphics logarithmic graph was an important tool for representation. For the drawing of diagrams in logarithmic representation there einfachlogarithmisches paper or double logarithmic paper. The graphical representations on the computer facilities have used logarithmic scale simplified and the use of such paper is greatly reduced.
Mathematical modeling
With the transition to a new variable
Arise for some functions simplifications in the representation, and certain relationships are illustrated. Conversely, it can be concluded on the underlying function from a straight course in a sequence of measuring points with suitable split-axis.
A power law is in double logarithmic representation of the straight line
The special case of a right under 45 ° falling line ( with the same scales on both axes) has b = -1, ie inverse proportionality.
An exponential curve can be represented as a straight line in a logarithmic representation simply
A function of the form of a normal distribution ( Gaussian bell curve ) is in simple logarithmic representation to a parabola
A function of the form of a log-normal distribution is simply in a logarithmic representation to a normal distribution and in double logarithmic representation to a parabola
Other applications
For certain tasks it has become common to scale an axis as the logarithm of the logarithm (log (log ( y ))), for example, the vertical axis in the graph of oil viscosities according to Ubbelohde -Walther or in the Weibull probability paper. Here the doubly logarithmic axis is sometimes referred to as double - logarithmic.