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(28.) If a + a ̧x + a ̧x2 + &c. = b + b1x + b2x2 + &c. for every value of x, then

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(29.) Theory of logarithms, and logarithmic tables.

(E. 220—55; Bour. 209-24.)

Let log a represent the logarithm of a to any base;
loga, the logarithm of x to the base a.

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This is only a particular case of the following more general theorems, which may be demonstrated in a similar manner :

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whatever may be the form or value of u,, then

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Either of these equations may be taken as the definition of a logarithm: the former is most usually adopted; and the various properties are therefore given in the order in which they most naturally follow on that supposition.

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log x1+log x2+ log x, + &c. = log (x1. x. x3. &c.)
x3

2

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2

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It will be hereafter seen that unity has an infinite number of logarithms, of which the above is the only possible one.

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=0, 4342944819032518276511289. log, x. log,10=2, 3025850929940456840179914 &c.

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1

5 +

1

Log, #=log, (#1)+2 { 2 — + 3(22—1); + &c.},

12x 1

3(2x-1)3

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a

β

Y - &c.

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2

3

+ &c.}..

δ

+

(Lagrange, Calc. des Fonctions, Leçon 4me)

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