Applied Calculus Math 215 - University of Hawaii

20 CHAPTER 1. SOME BACKGROUND MATERIAL
1.3 The Exponential and Logarithm Functions
Previously you have encountered the expression a x ,whereais a positive
real number and x is a rational number. E.g.,
10 2 = 100, 10 1/2 = √ 10, and 10 −1 = 1
10
In particular, if x = n/m and n and m are natural numbers, then ax is
obtained by taking the n-th power of a and then the m-root of the result.
You may also say that y = am/n is the unique solution of the equation
y n = a m .
By convention, a 0 = 1. To handle negative exponents, one sets a −x =1/a x .
Exercise 18. Find exact values for
−2 1
4
2
3/2
3 −1/2
25 −3/2 .
Exercise 19. Use your calculator to find approximate values for
3 4.7
5 −.7
8 .1
.1 −.3 .
Until now you may not have learned about irrational (i.e., not rational)
exponentsasinexpressionslike10 π or 10 √ 2 . The numbers π and √ 2are
irrational. We like to give a meaning to the expression a x for any positive
number a and any real number x. A new idea is required which does not only
rely on arithmetic. First, recall what we have. If a>1(resp.,0