7.7 *Elementary Transcendental Functions 427Proof. Suppose f: JR ~ ( 0, oo) satisfies the differential equation f' = k f,
Vx ER Define F(x) = lnf(x). Then by Theorem 7.7.2 (d) and the chain rule,
F'( ) = f'(x) = kf(x) = k
x f(x) f(x) ·
By Corollary 6.4.5, this means there is a constant c such that Vx E JR,
F(x) = kx + c; i.e.,
lnf(x) = kx + c; i.e.,
f(x) = ekx+c = ecekx_
Finish by regarding ec as a constant, and evaluate it by letting x = 0. •GENERAL EXPONENTIAL AND LOGARITHM FUNCTIONS
Definition 7.7.13 (General Exponential Functions) Suppose a > 0.
Then Vx E JR, we define1 7
Remarks 7. 7.14 Let a > 0.
(a) Definition 7.7.13 yields the conventional meaning of ax when x is a
natural number, integer, or rational number, and is consistent with Definition
5.6.5.
(b) The "laws of exponents" given in Theorem 7.7.10 (a)-(d) hold for ax.
In addition, Definition 7.7.13 allows us to add the following laws of exponents
and logarithms: V x,y E JR, (ex)Y = exv, (ax)Y = axv, and In(ax) = xina.
(c) The function ax is differentiable everywhere, and d~ax =ax Ina.
(d) If a> 1, the function ax is strictly increasing on JR, with range (0, +oo),
and lim ax = +oo while lim ax = 0.
x~+oo x~-oo
(e) If 0 < a < 1, the function ax is strictly decreasing on JR, with range
(0, + oo), and lim ax = 0 while lim ax= +oo.
x~+oo x~-oo
Definition 7.7.15 (General Logarithmic Functions) Suppose a> 0 and
a =f. 1. Then Vx E JR, we define the function loga x to be the inverse of the
function ax.
Remarks 7.7.16 Suppose a> 0 and a =f. 1. Then
- For students who studied Section 5.6, this is a theorem rather than a definition.