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

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We need to find the solution for log(x+3)=3-log(x-3)

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

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

use log a + log b = log (a*b)

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

=> (x + 3)*(x - 3) = 10^3

=> x^2 - 9 = 1000

=> x^2 = 1009

=> x = sqrt 1009 only as the log of a negative number is not defined.

**The solution of the equation is x = sqrt 1009**

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.)**