Elementary arithmetic, with brief notices of its history... by Robert Potts.

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mensurable; and show how its value may be determined to any
degree of accuracy when a = 10.
3. Assuming logyz logy. logx,
thence shew that logb. loga = 1, and log,. logc. log,a = 1.
4. Shew that the exponental expressions a', xy, y" may be put
under the forms exl~gc, e lge exl~gey respectively.
5. If eClo~a = b. elobx, find an expression for b.
6. Prove the truth of the three following expressions:n^,ogm -= mn10ogar: log1^ _ ~l0g: logr,, 10g,..logan logz a log(, log^y
fbl gam:-'2a' ogan log log.Y log~ y
7. Verify the following equivalents:
(b)loga+log*b - aloga blb* alT21ogb
and (6)lo1a - logb = (b - l)loga + logb = aloga. b-lo06
8. If yq = zr, prove that p logya = _ loga.
9. Shew that logc{2plj)o} =logan. loga. logb. log1a.
XV.
1 1
1. Prove that logc lies between n (1 -a ) and n(a - 1), when
n is assumed of such a value that a' is very nearly equal to unity.
1 1 1 1
2. Prove I -- 1.2.. 1..3.. 7 +.... 36788....
3. Shew that the value of (1 + -) = 2.7182... when x is indefinitely increased.
4. Shew that loge (1 + x) is less than x for all positive values of x,
whether greater or less than unity.
5. If x be very small, shew that eX ==e(l + x) nearly.
6. Shew that log10 l -log,99 = e 5- very nearly.
7. Prove that logx=-n(xT1- 1) nearly, when n is very great;
and thence shew that loge = (x - 1). 2 2 2
'x + 1 x4 +1 x + 1
8. Show that (1 + ) = fI -j-) nearly when n is large, and find
the next term of the series of which the function on the second side is
the commencement.
9. Prove that logo{ 1 + -- }is less than l --- and exemplify
9.10Jo
its truth when n is 3 and 4.
XVI.
1. If e'=y + V/1 + y, then y- (e -e-) and
V/ 1 + y2 -= (ex + e7-); and shew for what value of y, ex is less than 2.
2. If (a2- b)x = 1, shew that - =log(ac + b)
a + b 1-x log(a - b)