# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Log Rolling. Topic: Inequalities. Level: AIME. January 24th, 2007

Problem: Given reals , show that .

Solution: Recall the change-of-base identity for logs. We can rewrite the LHS as

.

Note, however that , so it remains to show that

,

which is true by Nesbitt’s Inequality. QED.

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Comment: A good exercise in using log identities and properties to achieve a relatively simple result. Once we made the change-of-base substitution, seeing the should clue you in to Nesbitt’s. That led to the inequality , which was easily proven given the conditions of the problem.

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Practice Problem: Given reals , find the best constant such that

.

### 3 Responses to “Log Rolling. Topic: Inequalities. Level: AIME.”

1. t0rajir0u Says:

The new problem is easier, I think. It becomes abc \ge (a + b + c)^k. At a = b = c = 2 we see we require k \le log_6 (8), and it’s pretty clear that if we increase a, b, c, the LHS increases faster than the RHS.

2. blue_giraffe Says:

we talked about logrolling in Health Science today, aka turning someone in bed with a draw sheet, used on people with spinal injuries or just out of spinal surgery. Thought it was a funny coincidence