solving nonlinear equation with logarithmic terms

Question

Let $0<a,b<1.$ How should one solve (for $x$) $$ x+a \log x= \log y -b \log \log y. $$ Please note the two opposite signs on $a,b.$ We can assume $y$ is very large and positive. Maybe the $W$ function is useful, but I am not an expert.

Even locating the solution in a manageable interval, whose two ends are in closed form would be good.

Answer 1

Let $z$ be your right-hand side. Then some manipulations (divide by $a$, exponentiate, divide by $a$ again) yields $$x+a \log x=z\implies (x/a)e^{x/a} = e^{z/a}/a\implies x/a = W(e^{z/a}/a)$$ where $W$ is the Lambert-W function. So $x=a W(e^{z/a}/a)$.

Answer 2

As @Semiclassical wrote,

$$x+a\log(x)=k \implies x=a W\left(\frac{1}{a}e^{\frac{k}{a}}\right)$$ If you cannot use Lambert function, you will need a numerical method and take into account the fact that $\forall t \geq e$ $$W(t) \geq\log (t)-\log (\log (t))+\frac 12\frac{\log (\log (t))}{ \log (t)}=f(t)$$ $$W(t) \leq \log (t)-\log (\log (t))+\frac e{e-1}\frac{\log (\log (t))}{ \log (t)}=g(t)$$

In the past, I minimized $$\Phi(a)=\int_e^{1000} \Big[a\, f(t)+(1-a) \,g(t)-W(t) \Big]^2\,dt$$ and obtained $a_{opt}=0.57958$ for which $\Phi(a_{opt})=0.134235$; for comparison, $\Phi(0)=35.6714$, $\Phi(0.5)=0.80423$ and $\Phi(1)=18.8333$.

Making numbers rational, a good approximation is $$W(t) \sim \log (t)-\log (\log (t))+\frac{11133}{11659}\frac{\log (\log (t))}{ \log (t)}$$ Over the considered range, the maximum error is $\sim 0.007$.

With such an estimate, Newton method would converge to high accuracy within a couple of iterations.

Edit

For illustration, let use use $y=10^{99}$, $b=\frac 12$ and $a=\pi$. This would give $t\sim 4.36783\times 10^{30}$, $W(t)\sim 66.3531$ and then $x\sim 208.454$ while the "exact" solution is $x \sim 208.466$.