The equation interested in is $\log_{10} p\times\log_{10} q=\log_{10} r$ where $p,q,r\in\mathbb N$ are natural numbers.
Here, I want not to consider some trivial solutions that make any one of logarithm terms integer, such as
$(p,q,r)=(10^2,10^3,10^6)$ or $(p,q,r)=(10,7,7)$
I guess there is no solution consist of natural numbers other than those trivial ones, but I'm having difficulty in verifying this..
So, could anyone give some answer to this problem?