4 ms·

I think the + and × are backwards from the way that would be helpful. But if you think it would work, try writing it out in detail.

• chii 5y ago
  let met try, for [a, b, c], and [d, e, f] 1. Take the log of all terms (allowed, because log keeps the sum monotonic): log(sqrt(a)) + log(sqrt(b)) + log(sqrt(c)) 2. pull out the sqrt from the log: 1/2 * log(a) + 1/2 * log(b) +1/2 * log(c) 3. factor out the 1/2: 1/2 * (log(a) + log(b) + log(c)) 4. sum of logs can be rewritten as a log of product: 1/2 * log (a * b * c) 5. compute log of (a * b * c), and halve it. Ditto with log of (d * e * f). This should give a number which is proportional to the original sum of sqrt.
  • jakear 5y ago
    > Take the log of all terms (allowed, because log keeps the sum monotonic) It seems you're employing a + b < c <=> log(a) + log(b) < log(c), which doesn't hold (consider 10, 10, and 20). (the real rule is a * b < c <=> log(a) + log(b) < log(c)), because log(a) + log(b) <=> log(a * b)
    • chii 5y ago
      ahh, that is where i tripped up! Good to see!
  • 3PS 5y ago
    This doesn't work. Even if you assume sqrt(a) + sqrt(b) > sqrt(c) + sqrt(d), that doesn't necessarily mean that log(sqrt(a)) + log(sqrt(b)) > log(sqrt(c)) + log(sqrt(d)). For example, let a = 1, b = 100, c = 25, d = 25. Then > sqrt(1) + sqrt(100) 11.0 > sqrt(25) + sqrt(25) 10.0 > log(sqrt(1)) + log(sqrt(100)) 2.302585092994046 > log(sqrt(25)) + log(sqrt(25)) 3.2188758248682006