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Question 53

The number of distinct integers $$n$$ for which $$\log_{\frac{1}{4}}({n^{2}-7n+11})>0$$,is

Solution

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For base of log in range $$1/4\in(0,1)$$ and $$\log_{1/4}(x)>0$$ is true only if 0<x<1. 

For integer n, $$x=n^2-7n+11$$ is an integer, so it cannot lie strictly between 0 and 1.

So, there is no integer value for which this inequality is satisfied.

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