Rule of 72

Yes, it is because the amount of time it takes to double your money is $\frac{\ln(2)}{\ln(1 + r)}$ (where $r$ is the interest rate), so if $\frac{c}{r} = \frac{\ln(2)}{\ln(1 + r)}$ , then a good approximation for $c$ where $0 \leq r \leq 0.16$ would be $c \approx 0.08\frac{\ln(2)}{\ln(1.08)} \approx 0.72$ However this breaks down if $r$ is significantly greater than $0.16$ ; for example, if $r = 0.32$ then $c \approx 0.80$ , so you might as well make up the rule of 80 at that point. The reason we use the rule of 72 is because interest and similar concepts almost always occur at rates between $0$ and $20$ in the real world.
Now here's the part you may find interesting if you were previously unaware of it! I don't really like the rule of 72 for any purpose other than super-fast approximation. Here's a slightly better (and slower) rule: the rule of three tenths.
Given $r$ , a better approximation for $c$ is $c \approx 0.69 + 0.003r$ This works because $\frac{d}{dr}\frac{r\ln(2)}{ln(1+r)} \approx 0.3$ when $0 \leq r \leq 2$ . Now let's try it out on a few well known interest problems:

Let $r = 8\%.$ By the rule of three tenths, $C \approx 69 + 0.3 * 8 \approx 72$ , so it takes roughly $72 / 8 = 9$ years to double your money. This is verified by the rule of 72.

Now let $r = 100\%.$ By the rule of three tenths, $C \approx 69 + 0.3 * 100 \approx 99$ . Therefore it will take $99 / 100 = 1$ year to double your money. This is verified by common sense (if you're making 100% per year, then it's doubling every year).

Here's my last example. If $r = 0\%$ , then common sense tells us we will never double our money. By the rule of three tenths, $C \approx 69$ . So it will take $69 / 0$ years to double our... wait, that's indeterminate, as predicted. (the rules of 72, 69, and 69.3 have this property as well)

Let me know if you have a better rule-of-thumb or disagree with mine!

EDIT: I'll also add that we primarily use the rule of 72 because it has 12 divisors, making it easy to divide.

I found out that the rule 72 works best with relatively low interest. For example the time it will take an investment to double is 12 years find the approximate interest rate implied by rule 72.Now suppose the doubling period is only 2 years .Is the approximation better or worse?

Right. $\ln 2 \approx 0.69$ so $t \approx 0.69 / 0.03 = 23$ .

We further approximate this as $t \approx 72 / r$ (with $r$ in percent), because 72 has several small factors that allow us to easily deal with $r = 2, 3, 4, 6, 8, \ldots$ .