first tier second tier third tier fourth tier

past issues

2004 2003 2002 2001 2000 1999 1998 1997 1996 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Welcome to NRICH.

Comparing a^b and b^a

---

---

By Franco Alberti (P2951) on Monday, September 25, 2000 - 08:06 pm:

I have a maths coursework IGCSE project and I need to find a way of proving which is bigger :
2 to the power of 3 or 3 to the power of 2.
I also have to investigate a formula to prove that for many other numbers. Please help me!

---

By Anonymous on Monday, November 20, 2000 - 05:55 pm:

Its a little advanced for GCSE, but for a,b real numbers greater than e = 2.71828... then we can prove that if a>b then a^b > b^a and vice vera.

Start with a^b > b^a then take logs to give
b ln a > a ln b, as both a,b >e >1
(ln a)/a > (ln b)/b
a < b because (ln x)/x is monotonic decreasing in the interval (e, infinity).
The argument works both ways.
Not sure this helps though!

---