Applied Calculus Math 215 - University of Hawaii

22 CHAPTER 1. SOME BACKGROUND MATERIAL
The theorem asserts that there is at least one real number 2 π which satisfies
these inequalities, and the uniqueness part asserts that there is only one
number with this property, making 2 π unique.
2
1.5
1
0.5
-1 -0.5 0.5 1 1.5
Figure 1.9: (1/2) x
The arithmetic properties of the exponential function, also called the
exponential laws, are collected in our next theorem. The theorem just says
that the exponential laws, which you previously learned for rational exponents,
also hold in the generality of our current discussion. You will derive
the exponential laws from the logarithm laws later on in this section as an
exercise.
Theorem 1.13 (Exponential Laws). For any positive real number a and
all real numbers x and y
a 0 = 1
a 1 = a
a x a y = a x+y
a x /a y = a x−y
(a x ) y = a xy
Some of the exponential laws can be obtained easily from the other ones.
The second one holds by definition. Assuming the third one, one may deduce
the first and third one. You are invited to carry out these deductions in the
following exercises.