Calculus: Analytic Geometry and Calculus, with Vectors

488 Exponential and logarithmic functions

Remark: The formulas (1) and (3) and (4) are examples of Stirling formulas for
log at and z!.
14 This problem requires that we capture the ideas used to obtain the number
y = e0.41631 690216

correct to 10 decimal places, it being assumed that a table giving e6.4163 correct

to 15 places and a desk calculator are available. The first step is to recognize

that y = AB, where -4 = ex, B = eh, x = 0.4163, and h = 0.00001 690216.

The number t4 is copied from the table correct to 12 or 15 places. The number

B is readily calculated to 12 or 15 places with the aid of the formula

hf2

which we shall learn about in a problem at the end of the next section. After

a few buttons have been pressed, the calculator will give the product y = AB

in a hurry.

15 This problem requires that we capture the ideas used to obtain the number

z = log 4.16316 90216

correct to 10 decimal places, it being assumed that a table giving log 4.1631 cor-

rect to 15 places and a desk calculator are available. The first step is to recognize

that

z = log 4.1631 + log

4.16316 90216

4.1631

and hence that z = I + B, where 14 = log x, B = log (1 + h), x = 4.1613, and

h _0.00006 90216

4.1631

The number -4 is copied from the table correct to 12 or 15 places. The number B

is readily calculated to 12 or 15 places with the aid of the formula

log (l + h) = h -

h2

+^33 -^4 + ...

The sum 14 + B can be obtained with a pencil, but it is safer to use the calculator.

16 Supposing that a desk calculator is available to perform additions, sub-

tractions, multiplications, and divisions, describe the steps by which the numbers

(a) 2=, (b) irr, (c) ex, (d) Ir6, (e) -,Vl-r

can be calculated correct to 12 decimal places. Hint: Take logarithms with

base e so that tables of values of e- and log x can be used. Partial ans.: Let w = a*

and obtain the formula log w = C, where C = AB, -4 = a, and B = log 7r.

Find a table giving 7r correct to 15D (that is, 15 decimal places). Then use the

method of Problem 15 to find B correct to 1SD. Then multiply to find C.

Then w = ec. Let C = n + x, where n is an integer and 0 5 x < 1, sow =

e"ez. Find e by multiplication or from a table. Find ez by the method of

Problem 14. Finally, multiply to get w.