Editing Common logarithm (section)

===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}}.]]