The binary logarithm is the logarithm to base 2.
The notation is sometimes used to denote this function in number theoretic literature. However, because Russian and German literature use the symbol to denote the base-10 logarithm and since this is the use recommended by the United States Department of Commerce (Taylor 1995, p. 33), this practice is discouraged. (To confuse matters even more, some German literature uses the notation to mean the binary logarithm.)
The binary logarithm is implemented in the Wolfram Language as Log[2, z] and Log2[z].
When information theoretic functions (like entropy) are computed using , the units of information are obtained in bits. When is used instead, the units of information are known as "nats."