This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1885 Excerpt: ... on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unity is its logarithm. The base in ordinary use is 10, and hence with this base log 10 = 1. Logarithms to this base are the only ones which will now be considered in their practical use. 3. Whatever the base ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1885 Excerpt: ... on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unity is its logarithm. The base in ordinary use is 10, and hence with this base log 10 = 1. Logarithms to this base are the only ones which will now be considered in their practical use. 3. Whatever the base be, the logarithm of unity is zero. For since, by Art. 1, log -+log b = log ab, let b = 1 and.." ab--a, .'. log a+log i = log a, .'. log 1=0. 4. Confining ourselves, as has been just premised, to 1o as the base, we have by Art. 1 log joo = log 102 = 2 log 10 = 2, log 10oo = log 108 = 3 log io = 3, log 10ooo = 4 log 10 = 4, and generally, if n be any positive integer, log 10" = n. Again, log = log i-log 10 =-log 10 log = log 1--log 10o---2, and generally, if n be any positive integer, log---=--n. 5 10" 5. The same number cannot have two different logarithms to the same base 10. For if it were possible that a had two different logarithms to the base 10, m and n suppose, then log ioa = log 10+log a-i+m, also = 1 +n. m and n cannot be different. 6. The same logarithm to the base 10 cannot belong to different numbers. For if it were possible that m were the logarithm of different numbers a and b, log = log a--log b--m--m = a.. % must be 1, b or a and b are not different. Characteristics of Logarithms. 7. It will be observed on inspection of the tables of logarithms to the base 10 that the logarithms of numbers are larger as the numbers are larger. Since log i is zero and log 10 is 1, the logarithms of all numbers between 1 and io are decimals between o and 1. Since log 10o--2, the logarithms of all numbers between 10 and 100 are unity with a decimal part attached. So the logarithms of all numbers between 100 and 1000 are 2 with a ...
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