Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Does $\log(x)$ stop at a certain value when x is infinite? Or is it also infinite?

I can see the graph go straighter and straighter in the horizontal direction, and I wonder if it will eventually be completely horizontal (i.e. gradient is equal to $0$).

Is that true or not?

share|cite|improve this question
$\log$ is base $10$? What is $\log(1000000000000000000000000000000)$? – Jonas Meyer Oct 30 '12 at 6:59
Assume it is base 10. I wonder if log(x) will stop at one point when the base is different. – user824294 Oct 30 '12 at 7:00
It doesn't matter what the base is. What is $\log_a(a^k)$? – Jonas Meyer Oct 30 '12 at 7:00
It is y=k iirc. – user824294 Oct 30 '12 at 7:02
user824294: $k$ could be negative or positive, but thinking of $k$ getting arbitrarily large is more relevant to your question. – Jonas Meyer Oct 30 '12 at 7:05
up vote 7 down vote accepted

While it is true that $\lim\limits_{x\to+\infty} \left(\frac{d\log x}{dx}\right) = \lim_{x\to+\infty} \frac 1x = 0$, i.e. the graph of $\log$ does get flatter and flatter as $x$ increases, we still have that $\log x \to +\infty$ as $x \to +\infty$.

An easy way to see this is to note that $\log$ is the inverse function to $\exp$ which is increasing and defined on all of $\mathbb R$ with $\lim\limits_{x\to+\infty} \exp x = +\infty$.

Things are of course not different if your logarithm is base $10$ as noted by the comments.

share|cite|improve this answer
one could note that the gradient of $\log$ is smaller than the gradient of any polynomial. – born Oct 30 '12 at 7:32

This is an informal graphical explanation. Reflect the graph of $ y = e^{x} $ about the line $ y = x $ so as to obtain the graph of $ y = \log(x) $. The statement that $ y = \log(x) $ has a horizontal asymptote is then seen to be mathematically equivalent to the statement that $ y = e^{x} $ has a vertical asymptote. However, the second statement cannot be true.

share|cite|improve this answer

The concept of limits can help here. The limit of Log(x) as x approaches huge values (infinity) is a huge undefined value and is not a constant value. Graphs may be misleading sometimes to extrapolate from.

share|cite|improve this answer

Considering natural logarithm you may just simply note that ,

$\ln e^x = x$, and $e^x → ∞$ is simply equivalent to $x→ ∞$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.