In
science and
engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a
logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
, the other on a
linear scale. It is useful for data with
exponential relationships, where one
variable covers a large range of values, or to zoom in and visualize that - what seems to be a straight line in the beginning - is in fact the slow start of a logarithmic curve that is about to spike and changes are much bigger than thought initially.
[(1) ]
(2)
All equations of the form
form straight lines when plotted semi-logarithmically, since taking logs of both sides gives
:
This is a line with slope
and
vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2:
:
A log–linear (sometimes log–lin) plot has the logarithmic scale on the ''y''-axis, and a
linear scale on the ''x''-axis; a linear-log (sometimes lin–log) is the opposite. The naming is ''output-input'' (''y''-''x''), the opposite order from (''x'', ''y'').
On a semi-log plot the spacing of the scale on the ''y''-axis (or ''x''-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the ''y'' values (or ''x'' values) to their log, and plotting the data on linear scales. A
log–log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.
Equations
The equation of a line on a linear-log plot, where the
abscissa axis is scaled logarithmically (with a logarithmic base of ''n''), would be
:
The equation for a line on a log–linear plot, with an
ordinate axis logarithmically scaled (with a logarithmic base of ''n''), would be:
:
:
Finding the function from the semi–log plot
Linear-log plot
On a linear-log plot, pick some ''fixed point'' (''x''
0, ''F''
0), where ''F''
0 is shorthand for ''F''(''x''
0), somewhere on the straight line in the above graph, and further some other ''arbitrary point'' (''x''
1, ''F''
1) on the same graph. The slope formula of the plot is:
:
which leads to
:
or
:
which means that
In other words, ''F'' is proportional to the logarithm of ''x'' times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points (''F''
0, ''x''
0) and (''F''
1, ''x''
1) will have the function:
:
log–linear plot
On a log–linear plot (logarithmic scale on the y-axis), pick some ''fixed point'' (''x''
0, ''F''
0), where ''F''
0 is shorthand for ''F''(''x''
0), somewhere on the straight line in the above graph, and further some other ''arbitrary point'' (''x''
1, ''F''
1) on the same graph. The slope formula of the plot is:
:
which leads to
:
Notice that ''n''
log''n''(''F''1) = ''F''
1. Therefore, the logs can be inverted to find:
:
or
:
This can be generalized for any point, instead of just ''F
1'':
:
Real-world examples
Phase diagram of water
In
physics and
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, propertie ...
, a plot of logarithm of pressure against temperature can be used to illustrate the various
phases of a substance, as in the following for
water:
2009 "swine flu" progression
While ten is the most common
base, there are times when other bases are more appropriate, as in this example:
Microbial growth
In
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditar ...
and
biological engineering
Biological engineering or
bioengineering is the application of principles of biology and the tools of engineering to create usable, tangible, economically-viable products. Biological engineering employs knowledge and expertise from a number o ...
, the change in numbers of
microbes due to
asexual reproduction
Asexual reproduction is a type of reproduction that does not involve the fusion of gametes or change in the number of chromosomes. The offspring that arise by asexual reproduction from either unicellular or multicellular organisms inherit the ...
and nutrient exhaustion is commonly illustrated by a semi-log plot. Time is usually the independent axis, with the logarithm of the number or mass of
bacteria
Bacteria (; singular: bacterium) are ubiquitous, mostly free-living organisms often consisting of one biological cell. They constitute a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria were am ...
or other microbe as the dependent variable. This forms a plot with four distinct phases, as shown below.
none, 500px, Bacterial growth curve
See also
*
Nomograph, more complicated graphs
*
Nonlinear regression#Transformation, for converting a nonlinear form to a semi-log form amenable to non-iterative calculation
*
Log–log plot
References
{{reflist
Charts
Technical drawing
Statistical charts and diagrams
Non-Newtonian calculus