Math CentralQuandaries & Queries


Question from Claire, a parent:

Suppose a, b are real numbers taking all positive values except 1, determine (without the use of calculators) whether there exists values of a and b such that log (base a) b + log (base b) a < 0


log(baseb) a, written logb a means the power to which you must raise b to get a.

For example, log2 8 = log2 23 = 3; i.e. you must raise 2 to the 3rd power to get 8.

Similarly, log2 2 is 1; log2 (1/2) = -1 since 1/2 = 2-1, etc.

Now what would log(½) 2 be? It's the power to which you must raise 1/2 to get 2.

But (½)-1 = 1/(½) = 2 thus log(½) 2 = -1.

Note that we've just seen log(½) 2 = log2 (½) = -1 so that we have a solution to your problem of a = 2 and b = ½. I think you can see that there would be infinitely many solutions.


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