How it has to be the argument of logarithmic function?Need example, please!

The argument of logarithmic function has to be positive,
for the logarithmic function to exist.

We know that the
domain of logarithmic function is (0 ; +infinite) and the codomain is R (real number
set).

f(x) = log
x

x>0

The argument
could be the variable itself, or an expression, like in this
case:

log (x^2 -6x + 5)

For
the logarithm function to exist, we'll impose the constraint to
argument:

x^2 -6x + 5 >
0

We'll find out the zeroes of the argument
first:

x1 = 1  and x2 = 5

The
argument is positive over the ranges:

 [(-infinite
1)U(5;+infinite)] intersected  (0;+infinite) =
(0;infinite)

Of course, there are cases when the argument
is positive for any real value of x, like in the case x^2 + x +
1.

x^2+x+1>0 for any value of
x.

Conclusion: the argument of logarithmic
function has to be positive, for the logarithmic function to
exist.