Q:

The difference between 'solution' and 'root'

I am wondering about the difference between the following demands:

Prove that P(x) has at least one root.

Prove that P(x) has at least one solution.

Are they the same?
The background to my question:

Let $c  \in \Bbb R$. Prove that the equation:
  $$\frac{1}{\ln x} - \frac{1}{x-1}$$
Has only one solution in $(0,1)$.

Here is what I have in mind:

Show that if $f'$ has no roots, then $f$ has one root at most.
Calculate the derivative. 
Show that when $\frac{1}{2}<x<1$, $f'(x)<0$
Show that $f'(1)=0$. This means that 0 is a local minimum point in $(0,1)$
Therefore, $f$ has at most one root.
Show from Intermediate value theorem that $f$ has a root in $(0,1)$.

A:

Yes, a root is basically the same as a solution. We often use the work "root" when we talk about polynomials.
Now, this doesn't mean that the words are interchangeable. We can say

The roots of the polynomial are ...

We can't say

The solutions to the polynomial are ...

Solutions are always to equations. Roots are for/of polynomials (functions).
You write that the equation
$$
\frac{1}{\ln(x)} - \frac{1}{x-1}
$$
has a solution, but what you have written isn't an equation because you don't have an equal sign. Instead you probably want to say that the equation
$$
\frac{1}{\ln(x)} - \frac{1}{x-1} =0 
$$
has a solution ...

Example: Let $f(x) = x^2 -3x + 2$. Then $f$ is a polynomial. We can say the following

$1$ and $2$ are the two roots of the polynomial $f$.
$1$ and $2$ are the two solutions to the equation $f(x) = 0$.

A:

Functions have roots, but equations have solutions.
So, we'd say that

the function $P$ has at least one root,

but not that "$P$ has at least one solution".
Conversely, we'd say that

the equation $P(x) = 0$ has at least one solution

but not that "$P(x) = 0$ has at least one root".
Of course the two notions are related, in that, by definition, $x_0$ is a root of $P$ iff $P(x_0) = 0$, or equivalently, if $x = x_0$ is a solution of $P(x) = 0$.

