What is the domain of definition of the function f given by f(x)=square root (ln^2x-3lnx+2) ?

The domain of a function f(x) is the set of values of x
for which the value of f(x) is real and defined.

For the
function f(x)=`sqrt((ln^2x-3lnx+2))` , firstly (ln^2x-3lnx+2) cannot be negative as the
square root of a negative number is a complex number.

This
gives: `(ln x)^2-3*lnx+2 >= 0`

`(ln x)^2 - 2*ln x -
ln x + 2 >= 0`

`ln x*(ln x - 2) - 1(ln x - 2)
>= 0`

`(ln x - 1)(ln x - 2) >=
0`

For this to be true either both of the terms in the
product should be positive or both of them should be
negative.

ln x - 1 >= 0 and ln x - 2 >=
0

ln x >= 1 and ln x >=
2

x >= e^1 and x >=
e^2

This is true for x >=
e^2

ln x - 1 < 0 and ln x - 2 <
0

ln x < 1 and ln x
<2

x < e and x <
e^2

As the logarithm is defined only for positive numbers 0
< x < e

The domain of the given function is
`(0, e)U[e^2, oo)`