# 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: $(13)$