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Negative logarithms

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. To facilitate this, a special notation, called bar notation, is used:

The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol n ¯ {\displaystyle {\bar {n}}} is read as "bar n ", so that 2 ¯.07918 {\displaystyle {\bar {2}}.07918} is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

* This step makes the mantissa between 0 and 1, so that its antilog (10) can be looked up.

The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten:

Note that the mantissa is common to all of the 5 × 10. This holds for any positive real number x {\displaystyle x} because

Since i is a constant, the mantissa comes from log 10 (x) {\displaystyle \log _{10}(x)}, which is constant for given x {\displaystyle x}. This allows a table of logarithms to include only one entry for each mantissa. In the example of 5 × 10, 0.698 970 (004 336 018...) will be listed once indexed by 5 (or 0.5, or 500, etc.).