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

23

7. Given logo12-= 1'0791812, log,01 121563 = '0498256,
log,oi 257915 = -0996512; find the value of {(1-44)-s- (1-44)-'}-12.
8. Solve the equation 10= 101 having given loge0 = 2*30258.
XIX.
1. If a, b, c be three consecutive numbers, prove that
logeb = 1-logec + logea 2aCC +- 1- )3 5(2a 1)&.
and explain the advantage of the use of this formula in calculating
the logarithms of prime numbers. Determine how many terms of the
series must be taken that the logarithms may be correct for seven
places of decimals, when the numbers consist of 2, 3, 4 or 7 digits.
2. Find the coefficient of x' in the expansions of (1 + x)".ex, and
of (1 - ). e-'.
3. If A2n-1 A22 be the coefficients of x2"1'-, x2, in the expansion of
log{(l + x)l-~-, according to the ascending powers of s;
shew that 2n (A, _ -1 A2,) = 1.
x - I os — ]
X 2 X4 x6
4. log,(l1 +x) + loge(l -x) 2 + ~ &c
1.2 3.4 5.6
5. Shew that x= e'-l~" arises from the elimination of y between
1 1
the equations y -el-g and z=el-'o:
and if (log z)' + (log y)r + (log z) = S,
prove that 3 + 2 + 3 + 4 +...... 0.
logran - (1- 1)2_ (1 + )-1)3 + (1 + -+_ -1)4-_
6. log,(n 2 -4 ( l)+ (L2 3
(( 1)2f-(1 - + )(a- 1)3+ - +L(?1 + )(a- 1)-&c.
XX.
1. Point out any advantages derived from the use of e as the base
for logarithms in analytical calculations, and of 10 as the base in arithmetical calculations.
2. Explain the terms, finite and infinite, definite and indefinite,
and shew that though each of the series -1 + + - +- &c., and
1 -+ - + - + &c. continued indefinitely, be infinite, their difference
is finite, and eclual to logg2.
/ 1 1 X2 XX
3. P rove that 1-kl +- lq-x 1+ + — '
4. Log, ). -_ x - +- 4-+.. ad infin.: hence calculate
logi,,4 to 5 places of decimals, having given logl0o = '43429.
5. Expand log,(x + 5x+ 6) in a series of descending powers of x.
6. Log (x+n)=:ogi)gn+ og(+ +log( 1.. +
+\ \ log +- + _.
- -1 +x'