First Digit Probability and Benford's Law

Abstract

The following work is written in easy language for college level students. It shows how the first digit probabilities of a group of continuous real-valued functions can be calculated. Thus, examples explaining how the probabilities are related to specific real-life situations, as well as the summary for all basic algebraic functions, were brought to the reader's attention. Besides, the Benford's formula was derived with no use of any additional guiding sources. Moreover, a comprehensive analysis of a group of certain discrete functions was performed by approximating the functions to the above-mentioned continuous ones, taking limits, and other methods. The work can be applied for calculating the first digit probabilities of more advanced functions as well while using the same approach. Furthermore, the technique can be useful while dealing with a large set of highly approximate numbers and a conclusion about their nature needs to be made.