Table of Contents

Last modified on March 22nd, 2024

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.

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.

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 = 2^{x} | 2^{0} = 1 | 2^{1} = 2 | 2^{2} = 4 | 2^{3} = 8 | 2^{4} = 16 | 2^{5} = 32 |

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

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).

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.

Now, plotting a function y = x^{3} 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 = x^{3} | 1^{3} = 1 | 2^{3} = 8 | 3^{3} = 27 | 4^{3} = 64 | 5^{3} = 125 | 10^{3} = 1000 |

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

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).

**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.