Verify if log equation have real root log(2x-5)=log(x^2+3)

To verify if the roots of the equation are real numbers,
we'll have to compute the roots. Before solving the equation, we'll impose the
constraints of existence of logarithms.

Since x^2+3 is
positive for any value of x, we'll set the only constraint for the given
equation:

2x -
5>0

2x>5

x>5/2

log
(2x-5) = log (x^2+3)

Since the bases are matching, we'll
use the one to one property:

2x - 5 = x^2 +
3

We'll move all terms to one
side:

x^2 + 3 - 2x + 5 =
0

We'll combine like
terms:

x^2 - 2x + 8 = 0

We'll
apply the quadratic formula:

x1 = [-b+sqrt(b^2 -
4ac)]/2a

x1 = [2+sqrt(4 -
32)]/2

Since sqrt (-28) is not a real value,
the equation has no real solutions.