[jkzxrsrm] powers of 10

FindRoot[10^x==10*x, {x,0.135}, WorkingPrecision->50]

10^0.13712885742386235368613621063 = 10*0.13712885742386235368613621063 = 1.3712885742386235368613621063

the other root is 10^1 = 10*1 .

the nontrivial root only exists when the base is greater than e, so there is no binary version.

4^(1/2) = 4*(1/2) = 2.  is this the only rational solution?