Common logarithm

To mitigate this ambiguity, the ISO 80000 specification recommends that log10(x) should be written lg(x), and loge(x) should be ln(x).

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.

Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well.

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.

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.

The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2.

When reading a number in bar notation out loud, the symbol

An alternative convention is to express the logarithm modulo 10, in which case with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.

[c] The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102: * This step makes the mantissa between 0 and 1, so that its antilog (10mantissa) can be looked up.

In 1616 and 1617, Briggs visited John Napier at Edinburgh, the inventor of what are now called natural (base-e) 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 "log(x)" when they meant log10(x).

Mathematicians, on the other hand, wrote "log(x)" when they meant loge(x) for the natural logarithm.

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.

The graph shows that log base ten of x rapidly approaches minus infinity as x approaches zero, but gradually rises to the value two as x approaches one hundred.
A graph of the common logarithm of numbers from 0.1 to 100
Page from a table of common logarithms. This page shows the logarithms for numbers from 1000 to 1509 to five decimal places. The complete table covers values up to 9999.
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 × 3 = 6 .
The logarithm keys ( log for base-10 and ln for base- e ) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation.