How to solve $4x-\log(x) = 0$

Question

I have a problem solving this equation: $4x-\log(x) = 0$. I can't seem to get this equation to a simpler form featuring $\log$s only or getting rid of the $\log$.

Is there a way to solve it without the Lambert-W function?

Answer 1

Applying a bit of algebra, we have $$4x-\log x=0$$ $$4x=\log x$$ $$x=\frac14\log x$$ Also note that $x$ is greater than $\log x$ at every point in $(0, \infty)$. Multiplying $\log x$ by $\frac14$ is only going to make it smaller. Therefore, there are no real solutions to this equation.

Answer 2

This is a transcendental equation that cannot be solved analytically. You can express it in terms of the Lambert W-function, as you suggested, or you can solve it numerically using methods like Newton's method.

Answer 3

No, the Lambert-W function is essentially the inverse of the exponential of your function, so if you could express the solution in terms of elementary functions then you could also do so for the Lambert-W. But it has been shown that that is impossible.

Answer 4

Since $e^{4x} > x$ for every $x \in \mathbb{R}$, it is impossible to have $e^{4x} = x$, so that the equation $4x = \log(x)$ has no real solution for $x > 0$, where this equation is defined.