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Explaining Logarithms, Exponents and Base Numbers With Plots
July 27, 2022

Table of Contents

Introduction

Logarithms, exponents and base numbers can be hard to think about at times. This blog visualizes these types of numbers using plots to show how they all relate to one another.

Be sure to check out a previous blog post explaining the math of logarithms and exponents!

Check out these other blogs on DSP math:

Plotting Exponentials

Figure 1 plots the exponential numbers 10^x, 2^x, and e^x. The base number determines how quickly the exponential number grows. The larger the base number the faster the growth.

Figure 1: Comparing exponential numbers 10^x, 2^x, e^x.
Figure 1: Comparing exponential numbers 10^x, 2^x, e^x.

Log10 Plot

Figure 2 shows the linear plot of x versus 10^x. In this plot the number 10^x appears exponential on the y-axis for increases in x. The base-10 logarithm returns the exponent of 10^x, which is x, which appears linearly on the plot.

Figure 2: A linear plot for x and 10^x.
Figure 2: A linear plot for x and 10^x.

Figure 3 is a logarithmic plot on the y-axis, referred to as a semilog plot. The logarithmic plot converts the numbers using log10() before plotting them. Therefore the y-axis shows the exponential 10^x as linear,

(1)   \begin{equation*}\text{log}_{10}\left( 10^x \right) = x\end{equation*}

and the number x as logarithmic,

(2)   \begin{equation*}\text{log}_{10}\left( x \right).\end{equation*}

Figure 3: A semilog plot on y for x and 10^x.
Figure 3: A semilog plot on y for x and 10^x.

Log2 Plot

Figure 4 shows the plot for a base-2 exponential. The figure shows a similar shape as Figure 1, just with a base number of 2.

Figure 4: A linear plot for x and 2^x.
Figure 4: A linear plot for x and 2^x.

Natural Log (Log e) Plot

Figure 4 shows the plot for a base-e exponential. The figure shows a similar shape as Figures 2 and 3, just with a base number of e.

Figure 4: A linear plot for x and e^x.

Conclusion

A linear plot will show an exponential number, say 10^x, as exponential and a linear number x as linear. However a logarithmic plot, or semilog plot, will show an exponential number as linear and a linear number as logarithmic.

Khan Academy also has a great video on  explaining logarithms you can check out here.

Check out these other blogs on DSP math:

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