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
- 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.
Linguistic Note
The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.
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: $(13)$