Logarithmic Inequality

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.