given number can be divided evenly by two. This definition gives rise to a function that coincides with the binary logarithm on the powers of two, but it is different for other integers, giving the 2-adic order rather than the logarithm. The modern form of a binary logarithm, applying to any number (not just powers of two) was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their applications in information theory and computer science became known. As part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy. Definition and properties The binary logarithm function may be defined as the inverse function to the power of two function, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse. Alternatively, it may be defined as , where is the natural logarithm, defined in any of its standard ways. Using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers. As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation: For more, see list of logarithmic identities. Notation In mathematics, the binary logarithm of a number is often written as . However, several other notations for this function have been used or proposed, especially in application areas. Some authors write the binary logarithm as , the notation listed in The Chicago Manual of Style. Donald Knuth credits this notation to a suggestion of Edward Reingold, but its use in both information theory and computer science dates to before Reingold was active. The binary logarithm has also been written as with a prior statement that the default base for the logarithm is . Another notation that is often used for the same function (especially in the German scientific literature) is , from Latin logarithmus dualis or logarithmus dyadis. The , ISO 31-11 and ISO 80000-2 standards recommend yet another notation, . According to these standards, should not be used for the binary logarithm, as it is instead reserved for the common logarithm . Applications Information theory The number of digits (bits) in the binary representation of a positive integer is the integral part of , i.e.