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In addition, an expression such as 5 the new de nition.
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which does not yet have any meaning, will be assigned a reasonable value by
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De nition: ax is the unique positive real number such that ln ax = x ln a (35.1)
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To see that this de nition makes sense, observe that the equation ln y = x ln a must have exactly one positive solution y for each real number x. This follows from the fact that ln y is an increasing function with domain (0, + ) and range the set of all real numbers. (See the graph of the ln function in Fig. 34-2.)
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PROPERTIES OF ax It is shown in Problem 35.4 that the function ax possesses all the standard properties of powers:
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(I) a0 = 1. (II) a1 = a. (III) au+v = au av . (IV) au v = (V) a v au . av 1 = v. a
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CHAP. 35]
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EXPONENTIAL FUNCTIONS
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(VI) (ab)x = ax bx . ax a x = x. b b u )v = auv . (VIII) (a (VII)
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35.3 THE FUNCTION ex For a particular choice of the positive real number a, the function ax becomes the inverse of the function ln x.
terminology Two functions f and g are inverses of each other if f undoes the effect of g, and g undoes the effect of f . In terms of compositions (Section 15.1), this means: f g(x) = x and g f (x) = x
De nition: Let e denote the unique positive real number such that ln e = 1 (35.2)
Since the range of ln x is the set of all real numbers, there must exist such a number e. It can be shown that e = 2.718 . . . . [See Problem 35.30(c).] Theorem 35.1: The functions ex and ln x are inverses of each other: ln ex = x Indeed, substituting e for a in (35.1), ln ex = x ln e = x 1 = x If we replace x in this result by ln x, ln eln x = ln x Hence, eln x = x [since ln u = ln v implies u = v by Property 10, Section 34.2] and eln x = x
Theorem 35.1 shows that the natural logarithm ln x is what one would call the logarithm to the base e ; that is, ln x is the power to which e has to be raised to obtain x: eln x = x. Theorem 35.2: ax = ex ln a . Thus, every exponential function ax is de nable in terms of the particular exponential function ex , which for this reason is often referred to as the exponential function. To see why Theorem 35.2 is true, notice that, by Theorem 35.1, ln ex ln a = x ln a But y = ax is the unique solution of the equation ln y = x ln a. Therefore, ex ln a = ax
EXPONENTIAL FUNCTIONS
[CHAP. 35
In Problem 35.9 it is shown that ex is differentiable and that: Theorem 35.3: Dx ex = ex . Thus, ex has the property of being its own derivative. All constant multiples Cex share this property, since Dx Cex = C Dx ex = Cex . Problem 35.28 shows that these are the only functions with this property. From Theorem 35.3, we have ex dx = ex + C Knowing the derivative of ex , we obtain from Theorem 35.2 Dx ax = Dx (ex ln a ) = ex ln a Dx x ln a =e This proves: Theorem 35.4: Dx ax = ln a ax ax dx = ax +C ln a
x ln a
(35.3)
[by the chain rule]
ln a = a ln a
or, in terms of antiderivatives,
(35.4)
We know that Dx x r = rx r 1 for any rational number r. Now the formula can be extended to arbitrary exponents. If, in Theorem 35.2, we replace x by r and a by x (thereby making x positive), we get x r = er ln x . Hence, Dx x r = Dx er ln x = er ln x Dx r ln x = er ln x r Thus, we have the following: Theorem 35.5: For any real number r and all positive x, Dx x r = rx r 1 1 x = xr r 1 x = rx r 1 [by the chain rule]