Why don’t logarithms have horizontal asymptotes?
We’re studying logarithms in my Honors Algebra II class, and when looking at a logarithmic function for the first time, a very astute student asked “does it have a horizontal asymptote.” And my quick response was a slightly dismissive “no.” But the student was insistent, and pointed out that the slope seems to keep getting smaller and smaller, so shouldn’t it eventually reach zero, and won’t that mean that there is an asymptote?
I like this question because, if I let it, it pulls me away from simply repeating the same information I was handed back during my math education, and really gets me thinking about the nature of the logarithm and the meaning of horizontal asymptotes.
What I “know”…
So here’s what I “know”—the logarithm is just the inverse of the exponential function, and the exponential function doesn’t have any vertical asymptotes—you can always exponentiate a larger number. Thus, it should be that when you invert this function to form the logarithm, there shouldn’t be any horizontal asymptotes. As you take logarithms of larger and larger numbers, the output of the function should continue to increase.
Let’s compare the logarithm to a function that we know does have an asymptote, namely, the inverse function. Just to make it a bit easier to visualize, I want to work with 1/x reflected over the horizontal axis and vertically shifted by 3,
In many ways this function looks similar to the logarithm—it is increasing, and as it grows, its slope decreases. But this particular function has a horizontal asymptote at .
What about the rate of change?
Things get more interesting when I look at the rates of change for these functions.
For the logarithm:
for the inverse function:
As we can see from the graphs, both slopes decrease as x increases, and the inverse decreases at a bigger rate. Does this difference explain this behavior?
It does seem to tell me that just because the rate of change of a function goes to zero at infinity can’t be proof of a horizontal asymptote. But could it be that any function whose rate of change goes to zero faster than a particular rate (say will have a horizontal asymptote?
I’m flummoxed. How would you explain to a student why a function whose rate of change is always decreasing never reaches a maximum value, particularly when every similar function he’s previously encountered like this does have a horizontal asymptote?