Two problems that I see regarding the lack of solution.
First, I dont think that the equation
has a solution.
Second, your kinda right about the solve function being inable. Solve works best for polynomial and algebraic equations. It can work with logs and exponents if they can be solved in an algebraic fashion.
Returns the answer
x -> 2+(2)^(1/2)
Returns a message
"The equations appear to involve the variables to be solved for in an essentially non-algebraic way."
Regarding the second part. Mathematica likes to put things into their basic elements. Input Log[a,x] and it will return Log[x]/Log[a]. Where Log is the natural log of course.