Logarithmic Inequality

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.

$$\log_Y{X}$$ > 1 (Since X > Y) $$\log_X{Y}$$ < 1 (Since Y< X) $$\log_Y{X}$$, $$\log_X{Y}$$ > 0 (Since X and Y are > 1)

All three statements are true.

Compare $$2^{\frac{1}{2}}$$, $$5^{\frac{1}{5}}$$, $$10^{\frac{1}{10}}$$ and $$20^{\frac{1}{20}}$$.

We can write log ($$2^{\frac{1}{2}}$$)= 1/2 log 2. Similarly other options will be 1/5 log 5 , 1/10 log 10, 1/20 log 20. 

LCM of 2,5,10 and 20 is 20, therefore it is as good as comparing $$2^{10}$$, $$5^4$$, $$10^2$$ and $$20^1$$. Therefore the largest number is $$2^{10}$$.

If a > 1 and $$log_{a}p > log_{a}q$$, then, p > q. Therefore, log ($$2^{\frac{1}{2}}$$) has the maximum value.