Question: How can I algebraically solve $n-\log(na)=b$ for $n$. I figured out a simple algorithm to get arbitrary close to the solution (estimate, then negate log term and use answer in the log term, then repeat), but I want to know how I can solve this exact.

Answer: 
If you cannot use Lambert function, the only way I can concieve is a root-finder method such as Newton. Provided a "good" guess, then the problem is quite simple. What you suggest would work but you have two manner for doing it. 

The first one is the one you suggest, that is to say repeating  

n = b + Log(n a)

The second one would be to repeat in the same spirit  

n = Exp[n - b] / a 

But both of same will be much slower than Newton iterative scheme which will write  

n = n (-1 + b + Log[a] + Log[n]) / (n -1)  

the rhs updating the lhs.
**Hint:** Use the [Lambert W-function](http://en.wikipedia.org/wiki/Lambert_W_function).