Which of the following is the largest number?
2^4000, 3^3000, 4^2500, 5^2000
**no calculator or logarithm tables.

Start by comparing 2^4000 and 4^2500 clearly the latter is 2^5000 and so larger, we can thus ignore 2^4000

Then noting that 3^3 is 27 and 5^2 is 25 we have

27^1000 and 25^1000, thus 3^3000 is larger than 5^2000

thus we only need to compare 4^2500 and 3^3000

looking at powers 4^2.5 is 4^2 * 4^0.5 or 16*2 = 32

3^3 is 27

so we have 3^3000 = 27^1000

and 4^2500 = 32^1000

So I reckon that 4^2500 is the largest, without the use of calculators or log tables.

David