What is the Solution? x ≥8lgx

Question

x ≥ 8lgx

I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.

Answer 1

If you want an analytical solution, you'll have to use the $W$ function.

See https://en.wikipedia.org/wiki/Lambert_W_function

Answer 2

Hint: We have $$x\geq8\log x$$ $$\frac{x}{8}\geq \log x$$ $$\log x\leq \frac{x}{8}$$ We know that $\log x$ is defined for all real values of $x$ such that $x\in(0, 1]$

Now, plot graphs of $y=\log x$ & straight line $y=\frac{x}{8}$

With the help of graphs, we find that both these graphs will intersect each other at a point say $x=a\ (>1)$ then the solution solution will be a set of real values $\color{red}{x\in(0, a]}$