Let's say that I start with the $\log_{10}7$, which is $0.84509804$, and I want to calculate $10^{0.84509804}$. $\ \ 0.84509804$ is close to $431/510$. Without access to anything electronic, it seems that I would have to find the $510^{th}$ root of $10^{431}$. I believe that would be an iterative process requiring calculating $X^{510}$ for each iteration. Is that how the table values were calculated prior to 1950? It seems unbelievably tedious.
I apologize for my lack of proper notation.