Math CentralQuandaries & Queries


Subject: large exponents
Name: nick
Who are you: Student

I am trying to figure out an extremely large number. It relates to the estimated number of bacterial divisions in 12775 generations of bacteria. Basically, I am trying to figure out approx. how many bacteria "live" during the course of an average human life cycle of 70 years. If a bacteria's life cycle is approx 2 days, that means there are 12775 generations of bacteria in one human lifetime. I know these numbers are inherently flawed because bacteria divide more than once per life cycle, but I just need a ballpark.

So, if each generation divided only once, that would mean I need the problem I need to solve is:

2^12775 or 2 to the power of 12,775.

I cannot find a calculator sold or even online that can compute such a large number. Scientific notation would be preferred, obviously.

Thanks very much for your help!!!


We have two responses to your question

Hi Nick.

Logarithms to the rescue:

212775 = 10b
ln (212775) = ln (10b)
12775 ln (2) = b ln (10)
b = 12775 ln (2) / ln (10)
b = 3845.65...


103845.65 = 100.65 x 103845 = 4.5 x 103845

So the "ballpark" is near 4.5 x 103845.

Hope this helps,
Stephen La Rocque.


Since the numbers are somewhat approximate I'll tell you how you can approximate your number. 210 is 1024 which is roughly 1000 = 103, thus 212775 is (210)1277.5 which is roughly (103)1277.5 or about 103833. Now, you should be cautious about using numbers like this as it is sometimes estimated that the number of atoms in the observable universe is of the order 1070. Since your number is of a completely different magnitude from this I'd say another look at your numbers is warranted.


Nick wrote back.

I wanted to thank Stephen and Penny for your help and timely response! It is much appreciated. While waiting for your reply, I came ~close to solving it by taking 21000, which my calc could do, and then multiplying that exponent, which was 301(1.07 x 10301) by 12.775 and came up with 3845 which was pretty darn close for my bass-ackwards math...

Anyway, I wanted to respond to penny's question about the number of atoms in the observable universe being 1070. I guess my rebuttal, more in the form of a question is, all matter/energy in the universe is neither created nor destroyed. So you could say it is recycled. So are my numbers all that crazy? Or am I crazy!?

You can look on the web at places such as:

for estimates of the number of atoms in the observable universe - google 'number of atoms in the universe'.

As for numbers of bacteria, that's not my game. The problem we have as humans is that we have no feel for large numbers. If you counted the number of different ways that you could line 20 children up for a school bus, and looked at a different arrangement every second, it would take 7 or 8 billion years to go through them all - roughly about the age of our universe (that's only about 1020 seconds). I'm not sure how you arrive at your numbers but certainly there could not be that many bacteria in 72 years - you might need to look at a population and consider a predator - prey model maybe, that is, not all bacteria can continue to divide repeatedly I suspect.


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