  Solutions to a wide variety of transcendental equations can be expressed in
terms of the Lambert $\mathrm{W}$ function. The $\mathrm{W}$ function,
occurring frequently in applications, is a non-elementary, but now standard
mathematical function implemented in all major technical computing systems. In
this work, we discuss some approximations of the two real branches,
$\mathrm{W}_0$ and $\mathrm{W}_{-1}$. On the one hand, we present some analytic
lower and upper bounds on $\mathrm{W}_0$ for large arguments that improve on
some earlier results in the literature. On the other hand, we analyze two
logarithmic recursions, one with linear, and the other with quadratic rate of
convergence. We propose suitable starting values for the recursion with
quadratic rate that ensure convergence on the whole domain of definition of
both real branches. We also provide a priori, simple, explicit and uniform
estimates on its convergence speed that enable guaranteed, high-precision
approximations of $\mathrm{W}_0$ and $\mathrm{W}_{-1}$ at any point. Finally,
as an application of the $\mathrm{W}_0$ function, we settle a conjecture about
the growth rate of the positive non-trivial solutions to the equation
$x^y=y^x$.
