Definition:General Logarithm/Binary

Definition

Logarithms base $2$ are becoming increasingly important in computer science.

They are often referred to as binary logarithms.

Also denoted as

A notation which is starting to take hold for the binary logarithm of $x$ is $\lg x$.

Some authors, particularly in the field of communication theory and cryptography, have been known to use the notation $\map \log x$ to denote $\log_2 x$, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Examples

Binary Logarithm: $\log_2 10$

The binary logarithm of $10$ is:

$\log_2 10 \approx 3 \cdotp 32192 \, 80948 \, 87362 \, 34787 \, 0319 \ldots$

Binary Logarithm: $\log_2 32$

The binary logarithm of $32$ is:

$\lg 32 = 5$

Also see

• Definition:Natural Logarithm
• Definition:Common Logarithm

• Results about logarithms can be found here.

Historical Note

The use of $\lg x$ instead of $\log_2 x$ to denote the binary logarithm was suggested by Edward M. Reingold, and adopted by Donald E. Knuth in his The Art of Computer Programming.

Sources

• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): Notation
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(13)$