Applications to Logarithms and Geometry

The integral definition of functions like C(x), S(x) of Fresnel makes them nearly as easy to use as elementary functions. It is possible to draw their graphs and tabulate values. You are asked to carry out an example or two of this on your problem set. To get used to using definite integrals and FTC2, we will discuss in detail the simplest integral that gives rise to a relatively new function, namely the logarithm.
Recall that ?
n+1
x
xndx =+ c
n +1
except when n = ?1. It follows that the antiderivative of 1/x is not a power, but something else. So let us define a function L(x) by ?
x
dt
L(x)=
t
1
(This function turns out to be the logarithm. But recall that our approach to the logarithm was fairly
involved. We first analyzed ax, and then defined the number e, and finally defined the logarithm as x
the inverse function to e. The direct approach using this integral formula will be easier.)
All the basic properties of L(x) follow directly from its definition. Note that L(x) is defined for 0 1
L?(x)=
x
Also, because we have started the integration with lower limit 1, we see that
? 1
dt
L(1)= =0
t
1
Thus L is increasing and crosses the x-axis at x = 1: L(x) < 0 for 0 0 for x> 1. Differentiating a second time,
L??(x)= ?1/x2
It follows that L is concave down.
The key property of L(x) (showing that it is, indeed, a logarithm) is that it converts multiplication into addition:
Claim 1. L(ab)= L(a)+ L(b)
Proof: By definition of L(ab) and L(a),
? ab dt ? a dt ? ab dt ? ab dt
L(ab)= =+= L(a)+
ttt t