calculate the solution of the equation log(x+3)=3-log(x-3)

We'll impose the constraints of existence of
logarithms:

x+3
>0

x>-3

x-3>0

x>3

The
interval of admissible solutions for the given equation is: (3 ;
+infinite).

Now, we'll sove the equation, adding log(x-3)
both sides:

log(x+3) + log(x-3) =
3

We'll apply the product
rule:

log (x+3)(x-3) = 3

We'll
recognize the difference of squares:

log (x^2 - 9) =
3

We'll take
antilogarithm:

x^2 - 9 =
10^3

x^2 - 9 = 1000

We'll add
9 both sides:

x^2 = 1009

x1 =
sqrt 1009

x1 = 31.764

x2 =
-31.764

Since the 2nd value of the root doesn't belong to
the range of admissible values, we'll reject
it.

The only valid solution of the equation
is:x = 31.764 (approx.)