Determin the values for x for the logarithm exists log(x+2)(x^2-4) .

We'll impose the constraints of existence of
logarithms:

1) x+2 >
0

2) x + 2 different from 1

3)
x^2 - 4 > 0

We'll solve the first
constraint:

1) x+2 >
0

We'll subtract 2 both
sides:

x > -2

2) x + 2
different from 1

We'll subtract 2 both
sides:

x different from -2+1 =
-1

3) x^2 - 4 > 0

We'll
re-write the difference of squares:

x^2 – 4 = (x - 2)(x +
2)

(x - 2)(x + 2)> 0

x
– 2>0 and x + 2>0

The expression is positive 
if x is in the interval (2 ; infinite)

x – 2<0 and x
+ 2<0

The expression is positive  if x is
in the interval (infinite ; -2)

The interval
of admissible values for the logarithm to be defined is
(infinite ; -2) U (2 ;
infinite).