Calculus: Analytic Geometry and Calculus, with Vectors

486 Exponential and logarithmic functions

For more than 300 years, logarithms with base 10 were systematically

and extensively used to make arithmetical calculations. Respectable

scientists of the present and future must know about them, but sub-

stantially all of the chores formerly done with the aid of tables Of 1093.0 X

are now done with slide rules and mechanical and electronic calculators

and computers of assorted shapes and sizes. It seems that the first

published table of logarithms with base e appeared in a 1618 edition of

the Napier tables, and that John Speidell used e as a base of exponentials

in a book published in 1620. Much of the present usefulness of a is

based upon work of Euler (1707-1783). Except for collectors of rare

books, tables of ex, ex, and log x are now vastly more valuable than

tables of logio x.

Problems 9.19

1 Find the values of a, x, and y that satisfy the equations

(a) 2z = 32, a6 = 32, 25 = y

(b) x = loge 32, 5 = log. 32, 5 = 1092 y

(c) 74 = y, 7x = 2401, a4 = 2401

(d) 4 = log? y, x = 1097 2401, 4 = log. 2401

(e) a$1000,103= y,1Ox=1000

(f) 3 = log. 1000, 3 = loglo y, x = loglo 1000

2 Practice the art of starting with the first of the equations

y = ax, log y = x log a = (log a)x, y = ekx,

taking logarithms (with base e) to obtain the second equation, and then using the

definition of logarithms to obtain the third equation where k = log a.

3 Practice the art of starting with the formula y = a= and writing

y = ax =(ek)x = ekx

where k is the exponent which we must put upon e to get a and hence k = log a.

4 Using the method of Problem 2 or the method of Problem 3, show that

(a) xx = e' 1- (x > 0)

(b) (1 +x)1Ix =ezlOg(I +X)

log (I +x)- log I

(c) f(x)o(x) = eo(o)1ogI(x) (f(x) > 0)

5 Show that akx = ekgx when k2 = k1 log a.

(^6) Sketch graphs of the equations y = 2x and y = loge x on the same sheet
of graph paper. Tell why the line having the equation y = x is (or is not) a
line of symmetry of the set consisting of the two graphs. Sketch a line which
appears to be tangent to the graph of y = 2z at the point (0,1), estimate the
coordinates of the point where this line meets the line having the equation y = x,
and use the results to obtain an estimate of the slope of the tangent. Finally,
modify the procedure to obtain an estimate of the slope of the line tangent to