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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?

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$\log$ 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.

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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.

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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.

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Considering natural logarithm you may just simply note that ,

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

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