# Bases Other than e and their Applications

## Among all choices for the base (b), b = e, 2, and 10 are particularly common for logarithms.

#### Key Points

• The major advantage of common logarithms (logarithms to base ten) is that they are easy to use for manual calculations in the decimal number system.

• The binary logarithm is often used in computer science and information theory because it is closely connected to the binary numeral system.

• Common logarithm is frequently written as "log(x)"; binary logarithm is frequently written "ld n" or "lg n".

#### Terms

• the exponent by which another fixed value, the base, must be raised to produce that number

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: $log_{10}(10x) = log_{10}(10)+ log_{10}(x) = 1 + log_{10}(x)$.Thus, log10(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 log10(x). For example, log10(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 log10(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 (log2 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 "ld n" or "lg n".

#### Key Term Glossary

binary
the bijective base-2 numeral system, which uses only the digits 0 and 1
##### Appears in these related concepts:
binary logarithm
The logarithm in base two.
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e
the base of the natural logarithm, 2.718281828459045…
##### Appears in these related concepts:
function
a relation in which each element of the domain is associated with exactly one element of the co-domain
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inverse
a function that undoes another function
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inverse function
a function that does exactly the opposite of another
##### Appears in these related concepts:
logarithm
the exponent by which another fixed value, the base, must be raised to produce that number
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natural logarithm
the logarithm in base e
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power
a measure of the rate of doing work or transferring energy