How do I solve for x? Do I need the Lambert W function?

Question

I need to solve the next equation x:

$d-x+yln[\frac{d}{x}]=b$

y, d, b, and x are all real, positive numbers.

How do I solve for x? Do use the lambert W function and if so how is that done?

Thanks!

Answer 1

Set $x=y t$ in order to get $$ t - \log\frac{d}{t y} = \frac{d-b}{y} $$ that is equivalent to: $$ t\cdot e^t = \frac{d}{y}\cdot e^{\frac{d-b}{y}} $$ so:

$$ x = y\cdot W\left(\frac{d}{y}\cdot e^{\frac{d-b}{y}}\right).$$

Answer 2

$$\begin{align*}d-x+y\ln\frac dx &= b\\ d +y\ln d-x-y\ln x &= b\\ d + y\ln \frac dy - \frac {yx}y - y\ln\frac xy &= b\\ d + y\ln \frac dy - b &= y\left(\frac xy + \ln\frac xy\right)\\ d + y\ln \frac dy - b &= y\ln\left(\frac xy\exp\frac xy\right)\\ \frac xy\exp\frac xy &= \exp\left(\frac{d - b}{y}+\ln\frac dy\right)\\ \frac xy\exp\frac xy &= \frac dy\exp\frac{d - b}{y}\\ \frac xy &= W\left(\frac dy\exp\frac{d - b}{y}\right)\\ x &= yW\left(\frac dy\exp\frac{d-b}y\right)\\ \end{align*}$$