Semi Log and Log Log Graphs

Semi-log graphs and log-log graphs are two different types of logarithmic graphs used for plotting statistical data with a wide range in magnitudes. 

Semi-log Graph

In a semi-log (or semi-logarithmic) graph, the y-axis is scaled logarithmically (the scales between the ticks on the graph are proportional to the logarithm of numbers), while the x-axis remains linear (the scales are evenly placed), making this graph a hybrid of the two scales.

However, semi-log graphs can also be represented with the logarithmic x-axis and the linear y-axis.

Interpretation

Now, plotting a function y = 2^x on a linear scale as well as on a semi-log scale, we get:

Selecting some points of the x-variables and evaluating the values of the function, we get

x	0	1	2	3	4	5
y = 2x	20 = 1	21 = 2	22 = 4	23 = 8	24 = 16	25 = 32

Now, using the values, we get the graphs as follows:

Observation

Since the y-axis of the semilog graph contains only the logarithms of positive numbers, the negative numbers are not included in this graph. 

It is useful for graphing exponential functions like y = ab^x, which produces a straight line with the slope of log(b) and the y-intercept of log(a). 

Log-log Graph

In a log-log (or double logarithmic) graph, both the x and y-axes use a logarithmic scale. 

It is used to create population charts, Zipf distributions, and air pressure measurements.

Interpretation

Now, plotting a function y = x³ on an ordinary axis (where the x and y axes are linear) as well as on a log-log scale, we get:

Selecting some points of the x-variables and evaluating the values of the function, we get

x	1	2	3	4	5	10
y = x3	13 = 1	23 = 8	33 = 27	43 = 64	53 = 125	103 = 1000

Now, using the values, we get the graphs as shown:

Observation

It is useful for determining power relationships between variables expressed in exponential form like y = ax^b, which produces a straight line with the slope ‘b’ and the y-intercept of log(a).

Solved Examples

E.g.1. Plot f(x) = ${\sqrt{x}}$ on a log-log graph.

Here, f(x) = ${\sqrt{x}}$

Selecting some points of the x-variables and evaluating the values of the function f(x), we get

x	1	4	9	16	25	100
f(x)	1	2	3	4	5	10

Now, using the values, we get the following log-log graph.

E.g.2. Plot f(x) = e^x on a semi-log graph.

Here, f(x) = e^x

Selecting some points of the x-variables and evaluating the values of the function f(x), we get

x	0	1	2	3	4
f(x)	1	2.72	7.39	20.1	54.6

Now, using the values, we get the following semi-log graph.