 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^{h}
< 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
0^{0} and division by zero are excluded.
First law (Additive):
b ^{p}b^{q} = b ^{p+q}
Second law (Subtractive):
b ^{p}/b^{q}
= b^{pq}
Third law (Multiplicative):
(b^{q})^{p}
= b ^{pq}
Fourth law (Distributive):
(ab)^{p} = a^{p}b^{p}
Fifth law (Distributive):
(a/b)^{p}
= a^{p}/b^{q}
 The function f is an exponential
function if
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 = Pe^{rt }
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}.
< Back
to Section 4





© 2011 Karl J. Smith. All rights reserved. 