# Common logarithm

The common logarithm.

In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use. It is indicated by log10(x), or sometimes Log(x) with a capital L (however, this notation is ambiguous since it can also mean the complex natural logarithmic multi-valued function). On calculators it is usually "log", but mathematicians usually mean natural logarithm rather than common logarithm when they write "log". To mitigate this ambiguity the ISO specification is that log10(x) should be lg (x) and loge(x) should be ln (x).

## Uses

Before the early 1970s, handheld electronic calculators were not yet in widespread use. Due to their utility in saving work in laborious multiplications and divisions with pen and paper, tables of base 10 logarithms were given in appendices of many books. Such a table of "common logarithms" gave the logarithm, often to 4 or 5 decimal places, of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001. There was only a need to include numbers between 1 and 10, since the logarithms of larger numbers can then be easily derived.

For example, the logarithm of 120 is given by:

$\log_{10}120=\log_{10}(10^2\times 1.2)=2+\log_{10}1.2\approx2+0.079181.$

The last number (0.079181)—the fractional part of the logarithm of 120, known as the mantissa of the common logarithm of 120—was found in the table.[note 1] The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, called the characteristic of the common logarithm of 120, is 2.

Numbers between (and excluding) 0 and 1 have negative logarithms. For example,

$\log_{10}0.012=\log_{10}(10^{-2}\times 1.2)=-2+\log_{10}1.2\approx-2+0.079181=-1.920819$

To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, a bar notation is used:

$\log_{10}0.012\approx-2+0.079181=\bar{2}.079181$

The bar over the characteristic indicates that it is negative whilst the mantissa remains positive. When reading a number in bar notation out loud, the symbol $\bar{n}$ is read as "bar n", so that $\bar{2}.079181$ is read as "bar 2 point 07918...".

Common logarithm, characteristic, and mantissa of powers of 10 times a number
number logarithm characteristic mantissa combined form
n (= 5 × 10i) log10(n) i (= floor(log10(n)) ) log10(n) − characteristic
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... 1.698 970...
0.000 005 −5.301 029... −6 0.698 970... 6.698 970...

Note that the mantissa is common to all of the 5×10i. This holds for any positive real number $x$ because:

$\log_{10}(x\times10^i)=\log_{10}(x)+\log_{10}(10^i)=\log_{10}(x)+i$.

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

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

$\begin{array}{rll} \text{As found above,} &\log_{10}0.012\approx\bar{2}.079181 \\ \text{Since}\;\;\log_{10}0.85&=\log_{10}(10^{-1}\times 8.5)=-1+\log_{10}8.5&\approx-1+0.929419=\bar{1}.929419\;, \\ \log_{10}(0.012\times 0.85) &=\log_{10}0.012+\log_{10}0.85 &\approx\bar{2}.079181+\bar{1}.929419 \\ &=(-2+0.079181)+(-1+0.929419) &=-(2+1)+(0.079181+0.929419) \\ &=-3+1.008600 &=-2+0.008600\;^* \\ &\approx\log_{10}(10^{-2})+\log_{10}(1.02) &=\log_{10}(0.01\times 1.02) \\ &=\log_{10}(0.0102) \end{array}$

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

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 2 x 3 = 6.

## History

Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17th-century British mathematician.

Because base 10 logarithms were most useful for computations, engineers generally simply wrote "log(x)" when they meant log10(x). Mathematicians, on the other hand, wrote "log(x)" when they meant loge(x) for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "ln(x)" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.

## Numeric value

The numerical value for logarithm to the base 10 can be calculated with the following identity.

$\log_{10}(x) = \frac{\ln(x)}{\ln(10)} \qquad \text{ or } \qquad \log_{10}(x) = \frac{\log_2(x)}{\log_2(10)}$

as procedures exist for determining the numerical value for logarithm base e and logarithm base 2.