I've reduced a set of equations to $V_t\ln(\frac{I_c}{I_s}) = V_{cc} - I_c*1000$
I need to solve this for $I_c$; everything other than $I_c$ are constants. I don't even know where to begin; normally I would exponentiate both sides but then I have an exponent of $I_c$ and I've just moved my problem without actually solving it ...
So I suppose the more general question is: How can I solve an equation with both $ln(x)$ and $x$ in the equation?
There is no analytical solution in terms of elementary functions.
However, the solution can be expressed in terms of Lambert function (which is such that $x=W(x)\, e^{W(x)}$); applied to your case, this will lead to $$I_c=\frac {V_t}{1000} W(z)$$ using $$z=\frac {1000\,I_s}{V_t}\,e^{\frac{V_{cc}}{V_t}}$$
In the Wikipedia page, you will find nice approximations of this function.
In fact, any equation which can write $A+Bx+C\log(D+Ex)=0$ has solutions which can be expressed in terms of Lambert function.
Otherwise, only numerical methods such as Newton would solve the problem.