How Rational is a Logarithmic Function
If asked to identify the function in Figure 1, almost every reader would respond
that it is a logarithmic function. In contrast, if the same question was posed to students at
the end of a standard course in college algebra or precalculus, most students would likely
say that it is a rational function. This is not all that surprising, given the relative
emphasis placed on both kinds of functions in traditional versions of these courses – on
the order of a week analyzing rational functions of all types versus only a few minutes
discussing the graph of a logarithmic function.
Figure 1
2
0
-2
0
0.5
1
1.5
2
2.5
-4
-6
-8
-10
-12
-14
However, it turns out that the students’ likely response is the correct one in this
case. The graph in Figure 1 is that of the rational function
A( x  1)
R( x) 
,
x
which has a vertical asymptote at the origin and a real zero at x = 1. In this expression, A
is a constant whose significance will be discussed later.
This leads to the question of just how similar a logarithmic function actually is to
a rational function? or, alternatively, how well can a logarithmic function be
approximated by a rational function? Such a discussion, as seen below, can provide some
valuable opportunities to reinforce key ideas and information in the minds of students.
As a preliminary answer to the above questions, consider Figure 2 that shows the
graphs of y = ln x and y = R(x) with A = 1.4 on the interval [0, 2.5], where we see that the
two functions are quite close to one another other than near the left end of the interval. In
fact, Figure 3 shows the graph of the error function, y = ln x - R(x), and we observe that
the values of the error range from about -0.05 to 0.08 on the interval [0.5, 2.5]. This
means that the rational function would give at least one-decimal place accuracy in
approximating the logarithmic function on this interval.
1.5
0.8
0.7
1
0.6
0.5
0.5
0
0
-0.5
-1
0.5
1
1.5
2
2.5
0.4
0.3
0.2
0.1
-1.5
0
0
-2
0.5
1
1.5
2
2.5
-0.1
Clearly, the region where the two functions differ markedly is near the vertical
asymptote. For now, we do not explicitly indicate which function is which in Figure 2.
Instead, we use the behavior of each function to make this decision. Realize that, since
the exponential and logarithmic functions are inverses of each other, their graphs are
symmetric. In particular, the portion of the exponential function that is growing rapidly
toward infinity as x increases to the right corresponds to the portion of the logarithmic
function beyond x= 1 that is increasing, albeit extremely slowly, toward infinity in a
concave down manner. It is convenient to think of this from a slightly different
perspective, however. We tend to think of the height of the exponential function; instead,
think of the horizontal distance from the y-axis to points on the right side of the
exponential curve. Those distances grow, but ever more slowly, as one traces along the
exponential curve.
This corresponds to the usual behavior we associate with the
logarithmic function, which we think of as growing ever more slowly as x increases.
Perhaps more significant to the present discussion, however, is looking at what happens
where the logarithmic function “dives” down toward the y-axis; this corresponds to the
portion of the exponential function that is decaying slowly toward the negative x-axis as a
horizontal asymptote.
How does this relate to a rational function? We know that, for x near 0, the
expression for R(x) behaves like y = -A/x with A > 0, which can be thought of either as
the power function with power -1 or as the hyperbola xy = -A, which has one branch in
the fourth quadrant. Either way, the curve in the fourth quadrant is symmetric about the
line y = -x. We also know that the exponential function dies out much more quickly than
any power function does as x . Because of the symmetries involved, it follows that
the logarithmic function falls toward its vertical asymptote of x = 0 much more quickly
than the power function y = -A/x does as x  0 from the right. In turn, this means that
the logarithmic function falls much faster than the rational function does, and so it
becomes evident which graph in Figure 2 is which.
We next look at the significance of the constant A. Again from Figure 2, we see
that the two curves cross twice, once at x = 1 and again at some later point beyond which
the logarithmic function grows more slowly than the rational function. For instance, if
the two intersect at x = 2, then we have ln 2 = R(2) = A(2 – 1)/2 = ½A and so A = 2 ln 2 =
ln 4  1.3863. In general, if the intersection occurs at x, then
ln x 
A( x  1)
 1
 A 1   ,
x
 x
and so
x ln x ln x x

.
x 1 x 1
Having this expression available allows us to investigate how well (or poorly) the
logarithmic function and a rational function agree. If we move the intersection point
further to the right, the two agree reasonably well for a longer interval starting at x = 1, at
least up to a point. However, the cost of doing this is to increase the separation between
the two curves between 0 and 1. On the other hand, moving the point of intersection
closer to x = 1 results in an improved match between the two on (0, 1], but the close
agreement beyond x = 1 does not persist for as large an interval.
If any readers care to extend the above discussion to see if they can improve on
the agreement between a rational function and the logarithmic function, particularly
between 0 and 1, we suggest introducing an additional parameter and considering rational
functions of the form
A( x  1)
R( x) 
,
xp
for positive integers p, all of which have a vertical asymptote at the origin and a real zero
at x = 1.
A