INDETERMINATE COEFFICIENTS.

(28.) If a + a,x + 0,2+ &c. = b + b 2 + box + &c. for every value of x, then a = b, ay = = b

&c. aIf a + a, x + a, x2 + &c. 0, then

0 = a; = &c.

(W. 346; Bour. 187, 8.)

LOGARITHMS.

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

(E. 220–55; Bour. 209–24.)

Let log x represent the logarithm of w to any base;

logail, the logarithm of x to the base a.

logam a

log x + log y= log xy.

a If SQM-1.2m-1 = $mbm-1.2m=,

an-1 = bm-1. This is only a particular case of the following more general theorems, which

may be demonstrated in a similar manner : If Šmam- "P,U, = 8,6 whatever may be the form or value of Ur, then

m-1

P, Ur?

m-1

· log x - log y=log

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.

Log a=1. log 1=0. am=7*. logja.

loga a
logz @=

logab
logab.log, a=1.
log X, + log x, + log x, + &c. = log(x,.X,.wz. &c.) Y

y

B

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

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

+

B

1 + 2

205

log: 1

+

Y

+ 3

+ &c. 5

log.(1 – x)=-(

- 6 + + + &e.).

{i &c.}. log. **] =2++se}

Cora)' + 3) + ke} .

8

log, x = log, a + 2

- a

1 + ta

3

2 + (1-uys S-1

1

1
log, w=2

+
+

+ &c.
læ+1

3
+

5

+ 1

1 log, x=log. (x - 1) + 2 +

+
2x

1 3(2x - 1)
1 (00* — 1) (Qc" - 1)3
log: x=n

&c.
1
2

3

1 log, x=n

+ 1

3 (Lagrange, Calc. des Fonctions, Leçon 4me.) 1

22n-1 231 - 1
log.

+
+

+ &c.
n 100n + 1 2 (27" + 1) 360+ 1)S

င်

4-0

+ &c

&c.}

&c.}

.

I log, x = log. (x - 1)+28m

(2m -- 1). (2x − 1)2 - 1