# How can I solve for x when it is in a natural log on the lhs and a polynomial on the rhs

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?

• The equation cannot be solved algebraically for $I_c$. – callculus Oct 19 '15 at 4:24
• Well! I guess my prof isn't likely to get mad at me for using wolfram alpha then, but how would(could) it be solved? – Daniel B. Oct 19 '15 at 4:26
• I don´t know, algebraically it is not possible. – callculus Oct 19 '15 at 4:41

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 fact, any equation which can write $A+Bx+C\log(D+Ex)=0$ has solutions which can be expressed in terms of Lambert function.