# Behavior of the natural log at large values of x

## Main Question or Discussion Point

Hello, I am fairly new here, so thank you in advance for your help. This is not a homework problem, just one of curiosity based on my limited knowledge of asymptotic expansions. I'm curious about how the function ln(x) or ln(1+x) behaves for large values of x. Plotting in Matlab, I see a linear behavior, but I am not certain as I cant seem to find an asymptotic expansion anywhere. Any help is appreciated, particularly with references.

D H
Staff Emeritus
The asymptotic behavior of log(x) is log(x). Log is a special thing, just as is exp(x). The exponential function grows faster than power function xa, no matter how large a is. This means that log(x) has to grow slower than any xa, no matter how small a is.

However, there is a lot one can say about log(x+1) for large x. Write x+1=x(1+1/x). Then log(x+1) = log(x(1+1/x)) = log(x) + log(1+1/x) ~ log(x) + 1/x.

thanks for the response....hmm, i cant seem to see a linear behavior in there, despite seeing a somewhat linear behavior when i plot it. Am i just visualizing something?

D H
Staff Emeritus
There is no linear behavior. The slope of log(x) approaches zero as x grows toward infinity, but it does so in a weird way: log (x) is of course unbounded.

There is no linear behavior. The slope of log(x) approaches zero as x grows toward infinity, but it does so in a weird way: log (x) is of course unbounded.
This was something that always puzzled me when I was taking calculus. As x increases, the graph of ln(x) "flattens out," but it flattens out so incredibly slowly that it still manages to cross every single horizontal line y=C. It's just such a counter-intuitive phenomenon. Can anyone shed any light into this mystery?

The same intuition that makes people think that ln(x) should have a horizontal asymptote would also make them think that e^x has a vertical asymptote somewhere. How is it that a function which becomes infinitely steep as x goes to infinity manages to still cross every vertical line x=C?

What mystery? It's just $$\lim_{x\rightarrow \infty} log(x) = \infty$$. It doesn't matter how "slow" it gets. It will get there "eventually".

Redbelly98
Staff Emeritus
Homework Helper
This was something that always puzzled me when I was taking calculus. As x increases, the graph of ln(x) "flattens out," but it flattens out so incredibly slowly that it still manages to cross every single horizontal line y=C. It's just such a counter-intuitive phenomenon. Can anyone shed any light into this mystery?
I guess I don't see what the mystery is, the part that I boldfaced actually sums it up very well. By the way, the square root function does this too -- though ln(x) does flatten out faster, and grows more slowly, than any positive-power function of x as DH said earlier.

But if you want a real mind-blower, look at a plot of ln(ln(x)), and try imagining that increasing without bound.

Hurkyl
Staff Emeritus
Gold Member
The growth of log(x) is closely related to the growth of the Harmonic series
1 + (1/2) + (1/3) + (1/4) + (1/5) + ...​