How to solve $\ln(x)-\ln(\ln(x))+\frac{\ln(\ln(x))}{\ln(x)}<4$?

Question

WolframAlpha gives $1<x<214.272$.

But on top of this specific solution, I am also interested in the general solution for:

$$\ln(x)-\ln(\ln(x))+\frac{\ln(\ln(x))}{\ln(x)}<K$$

With $K$ being a constant positive value.

Therefore, I would like to know how to approach that.

I tried substituting $\ln(x)$ with $z$ in order to reduce the problem to:

$$z-\ln(z)+\frac{\ln(z)}{z}<K$$

But I couldn't make any progress on that.

I also tried substituting $\ln(\ln(x))$ with $z$ in order to reduce the problem to:

$$e^z-z+\frac{z}{e^z}<K$$

But I couldn't make any progress on that either.

Any ideas how to approach this problem will be very much appreciated.

Thank you!

Answer 1

$$K>z-\ln(z)+\frac{\ln(z)}{z} = z\left(1- \frac{\ln(z)}{z} \right) + \frac{\ln(z)}{z}-1+1 = (z-1)\left(1- \frac{\ln(z)}{z} \right) +1$$ so $$(z-1)\left(1- \frac{\ln(z)}{z} \right) <K-1$$ Now taking $$1- \frac{\ln(z)}{z} <\frac{K-1}{z-1}$$ for $K-1>1$ you'll have solution $(0, x_0)$ for some $x_0>0$ which gives us intersection of two graphs. For $K=4$ $x_0>5$. Visually you can see this on desmos.com