It is known that $\log_{b}{a} = \frac{1}{\log_{a}{b}}$ So the given expression becomes $\log_{10}{ab}$ So we need to find $a, b$ such that $ab = 10^k$, where $k \leq 4$ We can easily find out all the pairs from here. (I'm too lazy to do it right now )

Salty_Titanium wrote: It is known that $\log_{b}{a} = \frac{1}{\log_{a}{b}}$ So the given expression becomes $\log_{10}{ab}$ So we need to find $a, b$ such that $ab = 10^k$, where $k \leq 4$ We can easily find out all the pairs from here. (I'm too lazy to do it right now ) They are (2,5), (5,2), (2,50), (4,25), (5,20), (10,10), (20,5), (25,4), (50,2), (10,100), (20,50), (25,40), (40,25), (50,20), (100,10), (100,100).