2.6 The Function bx
You have no idea what 2
is. You do know what 2n is for n an integer. You also know
what 2m∕n is for m,n integers. It is
a positive real number and r
number, define br ≡
. Does this definition contradict what we already
Proposition 2.6.1 For b > 0 and r a real number, define br ≡ exp
. Then if
= m∕n, for m,n integers,
that on the right meaning the positive nth root of bm.
Proof: From the above observation,
Therefore, taking nth
Recall that from Theorem 1.10.2
is a unique positive nth
root of a positive number so everything makes sense here.
This is a very important observation because it shows that if we define br as
there is no contradiction between this definition and what was already
accepted. It is not like what is done in religions when new doctrines contradict that
which was earlier accepted as true and everyone is supposed to choose to believe both
even though they contradict. Here we can with confidence make the following definition
real and b >
Definition 2.6.2 Let b > 0 and let r be a real number. Then br is defined
Proposition 2.6.3 The usual rules of exponents hold.
Proof: These properties follow directly from the definition.
Observation 2.6.4 Note that for e the number such that ln
and so from now on, I will use either ex or
they are exactly the same thing.
If you have x = f
in some interval, then
traces out a curve in the plane. These equations
called parametrizations of this curve. This will be discussed more later but it is
convenient to introduce the term now.
Definition 2.6.5 The hyperbolic functions, denoted as cosh
defined as follows:
The reason these are called hyperbolic functions is that
Thus if x = cosh
, then x2 − y2
= 1 so
parametrizes a hyperbola given by x2 −y2
= 1. The circular functions
so called because
. Thus if
parametrizes a circle.
Note that cosh
These are the even and odd parts of the function
t → et