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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. [b] Thus, log tables need only show the fractional part. Tables of common logarithms ...
For example, one can multiply a sine that is less than 0.5 by some power of two or ten to bring it into the range [0.5,1]. After finding that logarithm in the radical table, one adds the logarithm of the power of two or ten that was used (he gives a short table), to get the required logarithm. [1]: p. 36
For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Derivations also use the log definitions x = b log b (x ...
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
By repeated subtractions Napier calculated (1 − 10 −7) L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10 −5. Napier then calculated the products of these numbers with 10 7 (1 − 10 −5) L for L from 1 to 50, and did similarly with 0.9998 ≈ (1 − 10 −5) 20 and 0.9 ≈ 0.995 20. [32]
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The top left graph is linear in the X- and Y-axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y-axis of the bottom left graph, and the Y-axis ranges from 0.1 to 1000. The top right graph uses a log-10 scale for just the X-axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y-axis.
The line, inscribed on wood, was a single logarithmic scale going from 1 to 100. It had no sliding parts but by using a pair of dividers it was possible to multiply and divide numbers. [ note 3 ] The form with a single logarithmic scale eventually developed into such instruments as Fuller's cylindrical slide rule .