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

(In reply to My Solution - comments? by David Bate)

One of my solutions was basically the same: find the thousandth root of each:

2^4 = 16
3^3 = 27
4^2.5 = 32
5^2 = 25

Then since 4^2.5 is the largest, when all are raised to the 1000 power, this will be the largest also.

An alternative method, without strictly using calculator or log tables is to use a couple of logarithms known in one's head in approximate form, such as log 2 = .30103 (so that log 5 = 1 - .30103, or about .7).  In this case the only additional item one needs to remember is that the 3 was just slightly to the left of center on the C/D scales of a slide rule, indicating a common log just less than .5.  So the respective logs of the powers are about 1200, under 1500, about 1500, and 1400, and the third is again the highest.

Posted by Charlie on 2004-08-12 08:21:25