- Squeeze Theorem
Suppose b is a real number greater than 1. Then,
for any real number x, there is a unique real number b x. Moreover, if h and k are any two rational
numbers such that h < x < k , then
< b x < b k
- Extended Laws of Exponents
Let a and b be real numbers and let P
and Q be positive real numbers except that the form
00 and division by zero are excluded.
First law (Additive):
b pbq = b p+q
Second law (Subtractive):
Third law (Multiplicative):
= b pq
Fourth law (Distributive):
(ab)p = apbp
Fifth law (Distributive):
- The function f is an exponential
f (x) = b x
where b is a positive constant other than 1 and x
is any real number. The number x is called the exponent
and b is called the base.
- See the directory of curves for the graph
of the exponential function.
- Compound Interest: A = P(1 + r /n)nt
Continuous Compounding: A = Pert
where e is the natural base or Euler's number.
That is, as n increases without bound, the number
e is the irrational number that is the limiting value
of the formula (1 + 1/n)n.
to Section 4
© 2011 Karl J. Smith. All rights reserved.