General Power Functions
Defining General Power Functions
In previous lectures, we discussed the properties and derivatives of positive power functions and negative
power functions. Today, we discuss power functions in general.
A power function is a function of the form
f(x) = xa,
where ais any real number. We understand intuitively what it means to raise xto the power of a natural
number n: we just multiply ncopies of xtogether. We know what it means to raise xto the −npower:
just divide 1 by xn. Now we are asked to raise xto the power of a, where ais any real number, but what
does, say, 3πmean? It does not make sense to multiply πcopies of 3 together. We have been ignoring this
question since we defined exponential functions, but now we intend to give you an answer.
First, restrict xto the positive real numbers and consider the exponential function exand the natural
logarithmic function ln x. There is a great relationship built into exand ln x: if we compose these two
functions with each other in either order, the resulting function is x:
eln x=xand ln(ex) = x.
This relationship is built into the definition of the natural logarithmic function: the quantity ln xis defined
to be the number such that e raised to the power of that number is x. Thus we get the first equation above
automatically. The second equation comes from asking the question: to what number must we raise e to get
ex? Obviously, the answer to this question is x, and thus the natural logarithm of exis x. We say the the
functions exand lnxare inverse functions of each other: if we take xand first apply ex, then ln x, or the
other way around, then the result is the same as if we did nothing to xat all. In a sense, exand ln xcancel
each other out.
Still considering only positive real number x, consider the power function xa, where ais any real number.
We can use the fact that exand ln xare inverses of each other to rewrite the power function xa:
xa=¡eln x¢a= ealn x.
This last expression, ealn xis what we will use to define xafor positive real numbers x. That is, we define
the quantity xafor positive real numbers xand all real numbers ato be e raised to the power of the quantity
atimes the natural logarithm of x. In this way, we know how to compute, say, 3π, because we know what
it means to take the natural logarithm of 3, multiply it by π, and then raise e to the power of that number.
Thus, to the nearest thousandth, we have that
3π= eπln 3 = e3.142·1.099 = e3.451 = 31.544.
You may be thinking at this point that we merely shifted the problem from understanding how to raise
xto the power of ato understanding how to raise e to some power. How do we know how to calculate ea,
where ais any real number? There is an answer to this question, which is somewhat beyond the scope of
this course, but here it is anyway: there is another way to write ea. Specifically, we can write eaas
ea= 1 + a+a2
2+a3
6+a4
24 +···+an
n!+··· .
We will not tell you why eais equal to this infinite sum, this series as mathematicians would say, but it is
true, and you can verify it using any calculator. Specifically, you can show that
e = e1= 1 + 1 + 1
2+1
6+1
24 +· ·· +1
n!+··· ,
and, in fact, this is where the mysterious number e comes from. You do need to know all of this for this
course, but it does justify why we know how to raise e to any power a: we can compute the infinite sum
above (or, at least, approximate it) and get the same number.
So, at this point, we know how to define xawhen xis a positive real number. What about when xequals
0? To define 0a, we take the right limit of xaas xapproaches 0. There are three cases:
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