In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. It is also known as the decadic logarithm, the decimal logarithm and the Briggsian logarithm. The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm". Historically, the "common logarithm" was known by its Latin name logarithmus decimalis or logarithmus decadis.

The mathematical notation for using the common logarithm is , , or sometimes with a capital ; on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base ) 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, tabless 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

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, is2.

Negative logarithms

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 read as "bar 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

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 . This holds for any positive real number 

Since . This allows a table of logarithms to include only one entry for each mantissa. In the example of , 0.698970 (004336018...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

History

Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17thcentury British mathematician. In 1616 and 1617, Briggs visited John Napier at Edinburgh, the inventor of what are now called natural (base-) logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first chiliad of his logarithms.

Because base-10 logarithms were most useful for computations, engineers generally simply wrote "" when they meant . Mathematicians, on the other hand, wrote "" when they meant 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 "" 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.

To mitigate the ambiguity, the ISO 80000 specification recommends that should be , while should be written , which unfortunately is used for the base-2 logarithm by CLRS and Sedgwick and The Chicago Manual of Style.

Numeric value

The numerical value for logarithm to the base10 can be calculated with the following identities:

as procedures exist for determining the numerical value for logarithm base e (see ) and logarithm base 2 (see Algorithms for computing binary logarithms).

Derivative

The derivative of a logarithm with a base is such that (4 significant digits).

See also

• Binary logarithm

• Cologarithm

• Decibel

• Logarithmic scale

• Napierian logarithm

• Significand (also commonly called mantissa)

Notes

References

Bibliography