Suppose that I have to solve $x$ for$$\ln\Big(\frac{x}{x_0}\Big)=t(x+1)$$
This is my attempt, $$\frac{x}{x_0}=e^{t(x+1)}$$ $$=e^{tx+t}$$ $$x=x_0\cdot e^{tx+t}$$ $$\frac{x}{e^{tx}}=x_0e^t$$ $$xe^{-tx}=x_0e^t$$ $$-txe^{-tx}=-tx_0e^t$$ $$x=-\frac{W(-tx_0e^t)}{t}$$
Am I correct? Thanks in advance.