Editing Common logarithm (section)

==Mantissa and characteristic==
An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same [[fractional part]]. The fractional part is known as the '''mantissa'''.{{refn|group=lower-alpha|This use of the word ''mantissa'' stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text.{{citation needed|date=November 2024}} The word was introduced by [[Henry Briggs (mathematician)|Henry Briggs]].<ref>{{Cite book |last=Schwartzman |first=Steven |url=https://books.google.com/books?id=PsH2DwAAQBAJ |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms in English |date=1994-12-31 |publisher=American Mathematical Soc. |isbn=978-1-61444-501-2 |pages=131 |language=en}}</ref> The word "mantissa" is often used to describe the part of a [[floating-point]] number that represents its [[Significant figures|significant digits]], although "[[significand]]" was the term used for this by [[IEEE 754]], and may be preferred to avoid confusion with logarithm mantissas.}} Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999.

The integer part, called the '''characteristic''', can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation:

:<math>\log_{10}(120) = \log_{10}\left(10^2 \times 1.2\right) = 2 + \log_{10}(1.2) \approx 2 + 0.07918.</math>

The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is&nbsp;2.

===Negative logarithms===
Positive numbers less than 1 have negative logarithms. For example,

:<math>\log_{10}(0.012) = \log_{10}\left(10^{-2} \times 1.2\right) = -2 + \log_{10}(1.2) \approx -2 + 0.07918 = -1.92082.</math>

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:

:<math>\log_{10}(0.012) \approx \bar{2} + 0.07918 = -1.92082.</math>

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 <math>\bar{n}</math> is read as "bar {{Mvar|n}}", so that <math>\bar{2}.07918</math> is read as "bar 2 point 07918...".  An alternative convention is to express the logarithm modulo 10, in which case

:<math>\log_{10}(0.012) \approx 8.07918 \bmod 10,</math>

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.{{refn|group=lower-alpha|For example, {{cite journal
|year = 1825
|last = Bessel |first = F. W. |authorlink = Friedrich Bessel
|title = Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen
|journal = Astronomische Nachrichten
|volume = 331 |issue = 8 |pages = 852–861
|doi = 10.1002/asna.18260041601
|arxiv = 0908.1823
|bibcode = 1825AN......4..241B
|s2cid = 118630614 }}
gives (beginning of section 8) <math>\log b = 6.51335464</math>,
<math>\log e = 8.9054355</math>.
From the context, it is understood that
<math> b = 10^{6.51335464}</math>, the minor radius of the earth ellipsoid
in [[toise]] (a large number), whereas
<math> e = 10^{8.9054355-10}</math>, the eccentricity of the earth ellipsoid
(a small number).}}

The following example uses the bar notation to calculate 0.012 &times; 0.85 = 0.0102:

:<math>\begin{array}{rll}
 \text{As found above,}          &  \log_{10}(0.012) \approx\bar{2}.07918\\
 \text{Since}\;\;\log_{10}(0.85) &= \log_{10}\left(10^{-1}\times 8.5\right) = -1 + \log_{10}(8.5) &\approx -1 + 0.92942 = \bar{1}.92942\\
 \log_{10}(0.012 \times 0.85)    &= \log_{10}(0.012) + \log_{10}(0.85)                            &\approx \bar{2}.07918 + \bar{1}.92942\\
                                 &= (-2 + 0.07918) + (-1 + 0.92942)                               &= -(2 + 1) + (0.07918 + 0.92942)\\
                                 &= -3 + 1.00860                                                  &= -2 + 0.00860\;^*\\
                                 &\approx \log_{10}\left(10^{-2}\right) + \log_{10}(1.02)         &= \log_{10}(0.01 \times 1.02)\\
                                 &= \log_{10}(0.0102).
\end{array}</math>

<nowiki>*</nowiki> This step makes the mantissa between 0 and 1, so that its [[antilog]] (10{{sup|mantissa}}) 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:
{| class="wikitable" style="text-align:center;" border="1" cellpadding=5px
|+ Common logarithm, characteristic, and mantissa of powers of 10 times a number
! Number
! Logarithm
! Characteristic
! Mantissa
! Combined form
|-
! ''n'' = 5 × 10{{sup|''i''}}
! log{{sub|10}}(''n'')
! ''i'' = floor(log{{sub|10}}(''n''))
! log{{sub|10}}(''n'') − ''i''
!
|-
| 5 000 000
| 6.698 970...
| 6
| 0.698 970...
| 6.698 970...
|-
| 50
| 1.698 970...
| 1
| 0.698 970...
| 1.698 970...
|-
| 5
| 0.698 970...
| 0
| 0.698 970...
| 0.698 970...
|-
| 0.5
| −0.301 029...
| −1
| 0.698 970...
| {{overline|1}}.698 970...
|-
| 0.000 005
| −5.301 029...
| −6
| 0.698 970...
| {{overline|6}}.698 970...
|}

Note that the mantissa is common to all of the {{Math|5 {{times}} 10<sup>''i''</sup>}}. This holds for any positive [[real number|real number&nbsp;]]<math>x</math> because
:<math>\log_{10}\left(x \times10^i\right) = \log_{10}(x) + \log_{10}\left(10^i\right) = \log_{10}(x) + i.</math>

Since {{Mvar|i}} is a constant, the mantissa comes from <math>\log_{10}(x)</math>, which is constant for given <math>x</math>. This allows a [[Log table|table of logarithms]] to include only one entry for each mantissa. In the example of {{Math|5 {{times}} 10<sup>''i''</sup>}}, 0.698&nbsp;970 (004&nbsp;336&nbsp;018&nbsp;...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

[[Image:Slide rule example2.svg|thumb|500px|Numbers are placed on [[slide rule]] scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that {{Math|1=2 {{times}} 3 = 6}}.]]