Common logarithm

The mathematical notation for using the common logarithm is log(x), log₁₀(x), or sometimes Log(x) with a capital L; on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log", since the natural logarithm is – contrary to what the name of the common logarithm implies – the most commonly used logarithm in pure math.

Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive, and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. For the history of such tables, see log table.

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. 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: $${\displaystyle {\log }_{10}(120)={\log }_{10}\left({10}^2\times 1.2\right)=2+{\log }_{10}(1.2)\approx 2+\mathrm{0.07918.}}$$

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 2.

By applying this reasoning it can be seen that ${\displaystyle {\log }_{10}(120)=2.07918}$, ${\displaystyle {\log }_{10}(12)=1.07918}$, and ${\displaystyle {\log }_{10}(1.2)=\mathrm{0.07918.}}$

Negative logarithms

Positive numbers less than 1 have negative logarithms. For example, $${\displaystyle {\log }_{10}(0.012)={\log }_{10}\left({10}^{-2}\times 1.2\right)=-2+{\log }_{10}(1.2)\approx -2+0.07918=-\mathrm{1.92082.}}$$

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: $${\displaystyle {\log }_{10}(0.012)\approx \overline{2}+0.07918=-\mathrm{1.92082.}}$$

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 ${\displaystyle \overline{n}}$ is read as "bar n", so that ${\displaystyle \overline{2}.07918}$ is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case $${\displaystyle {\log }_{10}(0.012)\approx 8.07918mod10,}$$ 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: $${\displaystyle \begin{array}{rl}\text{As found above,}& {\log }_{10}(0.012)\approx \overline{2}.07918\\ \text{Since}{\log }_{10}(0.85)& ={\log }_{10}\left({10}^{-1}\times 8.5\right)=-1+{\log }_{10}(8.5)& \approx -1+0.92942=\overline{1}.92942\\ {\log }_{10}(0.012\times 0.85)& ={\log }_{10}(0.012)+{\log }_{10}(0.85)& \approx \overline{2}.07918+\overline{1}.92942\\ & =(-2+0.07918)+(-1+0.92942)& =-(2+1)+(0.07918+0.92942)\\ & =-3+1.00860& =-2+0.00860{}^{\ast }\\ & \approx {\log }_{10}\left({10}^{-2}\right)+{\log }_{10}(1.02)& ={\log }_{10}(0.01\times 1.02)\\ & ={\log }_{10}(0.0102).\end{array}}$$

* This step makes the mantissa between 0 and 1, so that its antilog (10^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:

Note that the mantissa is common to all of the 5 × 10ⁱ. This holds for any positive real number ${\displaystyle x\times {10}^i}$ because $${\displaystyle {\log }_{10}\left(x\times {10}^i\right)={\log }_{10}(x)+{\log }_{10}\left({10}^i\right)={\log }_{10}(x)+i.}$$

Since i is a constant, the mantissa comes from ${\displaystyle {\log }_{10}(x)}$, which is constant for given ${\displaystyle x}$. 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.).