Bases Other than e and their Applications

Among all choices for the base $b$, three are particularly common for logarithms. These are $b = 10$ (common logarithm); $b = e$ (natural logarithm), and $b = 2$ (binary logarithm). 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: 

$\log_{10}(10x) = \log_{10}(10)+ \log_{10}(x) = 1 + \log_{10}(x)$

Thus, $\log_{10}(x)$ 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-ten logarithms were given in appendices of many books. Such a table of "common logarithms" gave the logarithm—often to four or five 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-ten 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 $n \Rightarrow 2^n$. The binary logarithm of $n$ is the power to which the number $2$ must be raised to obtain the value $n$. This makes the binary logarithm useful for anything involving powers of $2$ (i.e., doubling). For example, the binary logarithm of $1$ is $0$, the binary logarithm of $2$ is 1, the binary logarithm of $4$ is $2$, the binary logarithm of $8$ is $3$, the binary logarithm of $16$ is $4$, and the binary logarithm of $32$ is $5$.

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 "$\text{ld}\, n$" or "$\lg n$".