ln(-x)=ln(x^2-6)How to find out the value of x in the above problem..please help

Since the bases are matching, we'll apply one to one rule
of logarithms:

-x = x^2 -
6

We'll use the symmetrical property and we'll
have:

x^2 - 6 = -x

We'll shift
-x to the left:

x^2 + x - 6 =
0

We'll apply quadratic
formula:

x1 =
[-1+sqrt(1+24)]/2

x1 =
(-1+5)/2

x1 = -2

x2 =
-3

Now, we'll impose the constraints of existence of
logarithms:

x < 0

x^2 -
6 > 0

The expression is positive if x belongs to the
intervals: (-infinite;-sqrt6)U(+sqrt6 ; +infinite)

The
common interval of admissible values for x is
(-infinite;-sqrt6).

Since x = -2 doesn't
belong to this interval, the only solution of the equation is x =
-3.