Calculus: Analytic Geometry and Calculus, with Vectors

484 Exponential and logarithmic functions

the procedure by letting r, s, t, and u be rational numbers for which r

x5sandt<y<=u. Then

aT <_ az < a8 and (aT), (ax)v (all)u,

so

art = (ax)v S a°u.

Taking limits as rt -+ xy and su -* xy gives axu <_ (ax)v S axe and shows

that (ax)v = axv.

Our basic theory of exponents enables us to give the basic theory of

logarithms in a few lines. With the aid of the intermediate-value

theorem, we see that to each positive number x there corresponds exactly

one number y such that ay = x. This number y, an exponent, is called

the logarithm with base a of x and is denoted by logo x, so that the formulas

(9.181) ay = x, y = log. X, a'°B°x = x

are equivalent. The logarithmic function is the inverse of the expo-

Y

Figure 9.182

x

nential function, and its graph is shown in Figure 9.182. The funda-

mental formulas

(9.183) log. xy = log. x + log. y, log. xv = y log. x

are proved by setting u = log. x, v = log,, y, x = au, and y = a'°, so that

xy = auav = au-M

and

log.xy = u+e = log. x+log.y.

Moreover, xv = (au)y = auv, so

log. xv = uy = y log. X.

When a, b, and x are positive numbers for which a > 1 and b > 1,

we can equate the logarithms with base b of the members of the identity

x = al-0 to obtain the first of the formulas

(9.184) logo x = (log. x)(logb a), 1 = (loga b) (109b a),