How to formulate that an equation be shown to have no solutions? Is there any general way to formulate the statement that an equation has no solution?
For example:

Prove that this equation has no solution: $$x^{1/\log x}=5$$ 


N.B. Do not answer with a proof of the example.
 A: How about saying that the solution set is the empty set?
A: Just say that $x$ doesn't exist:
$$
\nexists x \in \mathbb{C}\left[x^\left(1\over \log x\right) = 5\right]
$$
Slightly more formally:
$$
\forall x \left[x\in \mathbb{C} \implies x^\left(1\over \log x\right) \neq 5\right]
$$
A: In general if you are trying to solve some equation for $x$, and want to state there is no solution, you would say something alike the following:
There does not exist any $x$ such that $P(x)$ holds (or is true), where $P(x)$ is some truth statement. This can be expressed as
$$\not\exists x:P(x)$$
Read as "there does not exist any $x$ such that $P(x)$."
If you want to be more specific, you can state some domain of $x$, i.e.
$$\not\exists x\in A:P(x).$$
You would do this if you, for example, want to express that there is no real solution, i.e. if $A = \mathbb{R}$.

For your example
$$x^{\frac{1}{\log x}} = e^{\log\left( x^{\frac{1}{\log x}}\right)} = e^{\frac{1}{\log x}\log x} = e, $$
hence there is no solution to the given equation.
Let $P(x)$ be the truth-value of the equation $x^{\frac{1}{\log x}} = 5$ for a given $x$, then you could for example write
$$\not\exists x\in\mathbb{R}: P(x).$$
That is
$$\fbox{$\not\exists x\in\mathbb{R}: x^{\frac{1}{\log x}} = 5$} $$
A: The left hand side equals $e$ , so you have given a task to solve $ e =5. $
First inspect what is given and the basic logic of what you are asking, whether a set of variables can be combined in a functional relation to equal a constant.
The statement is not true, it is not even a valid equation even though $x's$ appear twice on the left hand side.
It is not an equation with any unknown, does not or need not have any solution real or complex.
It is not even an equation. By an equation we can imagine a set of scales where the the weight of items to be measured is in the right pan and a standard measure of weight in the left.
For it to be solvable left hand side minus right hand side should be a function of the variable. Else it is an empty set.
For a complicated real function either tabulate a few values or plot them in a graph with respect to the variable. The plot should necessarily go through zero for a real solution when a maximum or maximum exists.
For a real function $ y = f(x) $ the condition to have complex roots is 
either 
$$y^{'} =0, y> 0,  y^{''} > 0, $$
in the neighborhood of a real local minimum.
or
$$ y^{'} =0, y< 0, y^{''} < 0 $$ 
in the neighborhood of a real local minimum.
