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The base of the logarithm only affects a scaling factor and, consequently, the units in which the measured information content is expressed. If the logarithm is base 2, the measure of information is expressed in units of shannons or more often simply "bits" (a bit in other contexts is rather defined as a "binary digit", whose average ...
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.
For example, the pH of a 0.01 in moles per litreM solution of hydrochloric acid (HCl) is equal to 2 (pH = −log 10 (0.01)), while the pOH of a 0.01 M solution of sodium hydroxide (NaOH) is equal to 2 (pOH = −log 10 (0.01)), which corresponds to a pH of about 12.
The logarithm is denoted "log b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} .
An LNS can be considered as a floating-point number with the significand being always equal to 1 and a non-integer exponent. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication, and division, respectively.
where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x
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The logarithm function is not defined for zero, so log probabilities can only represent non-zero probabilities. Since the logarithm of a number in (,) interval is negative, often the negative log probabilities are used. In that case the log probabilities in the following formulas would be inverted.