Among all choices for the base b, three are particularly common for logarithms. These are b = 10 (common logarithm; see , b = e (natural logarithm), and b = 2 (binary logarithm; see . In this atom we will focus on common and binary logarithms.

The major advantage of common logarithms (logarithms in base ten) is that they are easy to use for manual calculations in the decimal number system: _{10}(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log_{10}(x).
For example, log_{10}(1430) is approximately 3.15.
The next integer is 4, which is the number of digits of 1430.

Before the early 1970s, hand-held electronic calculators were not yet in widespread use. Due to their utility in saving work in laborious multiplications and divisions with pen and paper, tables of base 10 logarithms were given in appendices of many books. Such a table of "common logarithms" gave the logarithm--often to 4 or 5 decimal places-- of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001. There was only a need to include numbers between 1 and 10, since the logarithms of larger numbers were easily calculated.

Because base 10 logarithms were most useful for computations, engineers generally wrote "log(x)" when they meant log_{10}(x).
Mathematicians, on the other hand, wrote "log(x)" when they meant log_e(x) for the natural logarithm.
Today, both notations are found.
Since hand-held electronic calculators are designed by engineers rather than mathematicians, they customarily follow engineers' notation.

Binary logarithm (log_{2} n) is the logarithm in base 2.
It is the inverse function of

The binary logarithm is often used in computer science and information theory because it is closely connected to the binary numeral system. It is frequently written as "ld n" or "lg n".