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Jim Lamb ers Math 2B Fall Quarter 2004- Le ture 15 Notes
These notes orresp ond to Se tion 7.4* in the text.
General Logarithmi and Exp onential Fun tions
General Exp onential Fun tions
The natural logarithmi and exp onential fun tions an b e used to de ne exp onentiation using any base a > 0. Sin e a an b e written as eln^ a^ using the an ellation equations, it follows that
ax^ =
eln^ a^
(^) x = ex^ ln^ a^ (1)
sin e (ex^ )y^ = exy^ for any x and y. We thus have the following de nition.
De nition 1 (General Exponential Fun tion) Let a be a positive real number. The exp onential fun tion with base a, denoted by ax^ , is de ned by
ax^ = ex^ ln^ a^ ; (2)
where x is any real number.
Example 1 Write
p 2
p 3 using the natural exp onential and logarithmi fun tions.
Solution Sin e the general exp onential fun tion ax^ is de ned by ax^ = ex^ ln^ a^ , it follows that we an set a =
p 2 and x =
p 3 and obtain (^) p 2
p 3 = e
p 3 ln p 2 : (3)
By the laws of logarithms, we an eliminate one square ro ot:
p 2
p 3 = e(
p 3 ln 2)= 2 : (4)
2
This de nition an b e used to easily prove the well-known laws of exp onentiation:
ax+y^ = ax^ ay^ (5) ax y^ = ax^ =ay^ (6) (ax^ )y^ = axy^ (7) (ab)x^ = ax^ bx^ (8)
It is not hard to see that these laws follow almost dire tly from the orresp onding laws for the natural exp onential fun tion. The ab ove de nition an also b e used to ompute the derivative of ax^ , using the Chain Rule. This leads to the following di erentiation rule and orresp onding integration rule.
Theorem 1 Let a be a positive real number that is not equal to one. Then
d dx (a
x (^) ) = ax (^) ln a (9)
and (^) Z ax^ dx = a
x ln a
+ C ; (10)
where C is an arbitrary onstant.
In your previous al ulus ourse, you learned the p ower rule for di erentiation, d dx
(xn^ ) = nxn ^1 ; (11)
where n is any integer. Using the de nition of exp onentiation, we an write xn^ = en^ ln^ x^ and use the Chain Rule to prove that the p ower rule a tually holds for any real numb er n.
General Logarithmi Fun tions
The natural logarithmi fun tion an also b e used to de ne the logarithmi fun tion with an arbitrary base a: Su h a de nition is ne essary sin e ertain bases, su h as a = 2 and a = 10, are frequently used in appli ation areas su h as omputer ar hite ture or numeri al analysis. To obtain a useful expression for the logarithm of a numb er x in base a, we use the exp onential fun tion with base a.
De nition 2 (General Logarithmi Fun tion) Let a be a positive real number. The logarithmi fun tion with base a, denoted by log (^) a x, is de ned by the relation
log (^) a x = y () ay^ = x; (12)
where x is a positive real number and y is a real number.
The left side of equation (12) is known as the logarithmi form, while the right side is alled the exponential form. Often, it is ne essary to onvert from one form to the other to solve an equation involving exp onential or logarithmi fun tions. Taking the natural logarithm of b oth sides of ay^ = x and rearranging algebrai ally, we obtain a more useful form of the de nition,
log (^) a x = ln^ x ln a
Sin e it is y that we want to nd, we must solve this equation for y. Exp onentiating b oth sides with base 2, we have (y + 3)^2 = 2 x^ : (19)
Taking the square ro ot of b oth sides yields
y + 3 = (2x^ )^1 =^2 = 2 x=^2 : (20)
We on lude that y = f ^1 (x) = 2 x=^2 3 : (21)
2
The Numb er e as a Limit
If f (x) = ln x, then f 0 (1) = 1. From the de nition of the derivative, it follows that
xlim! 0
ln(1 + x) ln 1 x =^1 :^ (22)
However, sin e ln 1 = 0 and x ln a = ln ax^ for any real x and a > 0, it follows that
xlim! 0 ln(1^ +^ x)^1 =x^ =^1 :^ (23)
Exp onentiating b oth sides yields the following alternative de nition of e,
xlim! 0 (1^ +^ x)^1 =x^ =^ e:^ (24) By making the substitution x = 1 =n, we obtain yet another alternative expression for e,
e = (^) nlim!
n
n : (25)
This expression is useful in determining the value of an investment after a given p erio d of time if interest is omp ounded ontinuously, rather than at regular intervals.
