Table of logarithms

Compared wit other bases, Brigg's logarithms had advantages which were a consequence of the agreement of the base with our number system. Therefore we will only consider here Brigg's logarithms. Some of these advantages are:

All other logarithms are irrational numbers. For example, 0 < log 7 < 1. If you assume that log 7 is a fraction, say p/q (p, q having no common factors and q > 1), then you must have

10^p/q = 7 or 7 = , which is irrational (cf. Irrational numbers)

Obviously, the integer 7 cannot equal an irrational number, whence the initial assumption was false. If the irrational numbers of the logarithms have four decimals, you speak of four-place logarithms.

Tasks: Given that log 7.21 = 0.8597, find log 72.1, log 721, etc.! Apply Rule I for logarithms:

You still have to combine the numbers on the right hand sides. You do that as far as addition is concerned and write log72.1 = 1.8597, etc. In contrast, subtractions, for example in log0.721 = 0.8597-1, is not carried out, but you compute with the differences.

The number after the decimal point is called mantissa, the other number characteristic, that is, every logarithm has a mantissa and a characteristic. Because the subtraction is not performed, you have the Rule:

Numbers with the same sequence of digits have the same mantissa.

In tables, you only find the mantissa, the characteristic is readily determined. For numbers > 1, you have

log 1 = 0, log 10 = 1, log 100 = 2, log 3000 = 3,

so that the logarithm of a number with one place in front of the decimal point (for example, 2.4) lies between 0 and 1, that is, it begins with 0, that of a number with two digits in front of the decimal point (for example, 24.6) begins with 1, etc., whence

If a number > 1 has n digits before the decimal point,
its characteristic is n -1.

Task: Find log 125!

A table of logarithms, a section of which is shown, is organized in such a manner that the log N of a given number N, for example 125, is found by locating to start with the first two digits, in this case 12, in the first vertical column and then the third digit in the labelled horizontal line, in this case 5. You thus find the mantissa 0969 and determine the characteristic 2 by the rules above, whence log 125 = 2.0969.

Examples: Find 1. log 16.2. By the table shown above, the mantissa is 2095, the characteristic, because the number has two digits in front of the decimal point, is 1, whence

Next, consider numbers < 1. From above, it follows that the logarithm of a number, which has 0 before the decimal point (that is, 0.···· -n), must lie between 0 and -1 and therefore has the characteristic n =-1, that of a number with one 0 after the decimal point has the characteristic n = -2, etc., whence

Hence follows for numbers between 0 and 1 the Rule:

Logarithms of numbers which start with 0...., also the logarithm starts with zero. The characteristic is minus the number of zeros before
the first non-zero digit.

Examples: Find log 0.172. The corresponding mantissa (cf. the table above) is 2355, the number of zeros is 1, whence

Since the table of logarithms above refers to numbers with three digits, you round off at first to three places, for example a = 43.6458 to 43.6. Similarly, if a given logarithm is not exactly in the table, you use the entry closest to the given value.

Examples:

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