Thermodynamic Entropy and Information

What is the math symbol "log" and "ln"?

    A logarithm (log) is a simple math operator that gives the exponent of the argument. For example 1000 = 10^₃ (10*10*10), so log₁₀(1000) = 3. Another way of saying this is that three decimal (base 10) digits are enough to specify 1000 states (000 to 999). Because bits (binary digits) are used in information theory and computer engineering, a binary (base 2) system is more convenient. For example 64,536 = 2¹⁶ so log₂(64,536) = 16. 16 bits are enough to specify 64,536 states.

    The logarithm of a number is just another way of specifying the size of that number. It appears frequently in statistical problems because the number of possible combinations of n objects is exponentially related to n. For example, the number of ways that 3 base 10 digits can be combined is 10³.

    The number system (base) that is used is a matter of preference. Humans count on their fingers, so base 10 is common. Computers only have many hands with only one finger, and it's either up or down, so base 2 is used. Another commonly used base is e = 2.718281...(because it makes some math problems easier) and a logarithm with this base is usually written as ln (the natural logarithm) instead of log_e.