Conversion of an implicit to an explicit funtion

Question

${1 \over a}({1 \over x} - {1 \over x_0}) - b \ln({x \over x_0}) = t$; where $a, x_0$ and $b$ are constants. Find an explicit function $t \mapsto x(t)$.

I was suggested the use of Lambert W function to get an explicit function x(t). Anyone familiar with it?

Answer 1

As expected, the inverse function involves the Lambert W function :

Answer 2

For any equation of the form $$a + b x + c \log(d + e x)=0$$ $x$ can be expressed in terms of Lambert function and the result is (I assume that all coefficients are non zero $$x=\frac{c W(z)}{b}-\frac{d}{e}$$ in which $$z=\frac{b e^{\frac{b d-a e}{c e}}}{c e}$$ In your case, start replacing $x=\frac{1}{y}$ which makes your expression very close to the one I wrote and you will obtain what JJacquelin answered.