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The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a [4] (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural".
Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors: = + , provided that b, x and y are all positive and b ≠ 1.
Positive numbers less than 1 have negative logarithms. For example, = = + + = To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa.
Since the common logarithm of a power of 10 is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For positive numbers less than 1, the characteristic makes the resulting logarithm negative, as required. [38]
If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.: = = = = (). The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used.
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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.
A negative means you aren’t able to, and positive means you are able to. They can’t tell you for certain whether or not you have, or had, COVID, according to Adalja.