SOLUTION: Prove that the equation: {{{ log( 2, x ) = 2^(-x)}}} had exactly one real solution within the interval (1,2).

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Question 1058942: Prove that the equation: +log%28+2%2C+x+%29+=+2%5E%28-x%29 had exactly one real solution within the interval (1,2).
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

I solved by the following reasoning:
Slope of log2(x) over (1,2) is positive and rising from 0 to 1 ( log2(1)=0, log2(2)=1 )
Slope of +2%5E%28-x%29++=+1%2F2%5Ex+ is negative and the function decreases over (1,2) from 1/2 to 1/4

So log2(1) < 1/2 and log2(2) > 1/4
That means the two functions cross on (1,2) and since they have slopes of opposite signs, they will only cross once on that interval.