find the exact value

Question

log₃(x-2)- log₃(x+2)= 2

Find the exact value for x

thanks

Answer 1

there is no solution:

By properties of exponents:

log_3[ (x-2)/(x+2) ] = 2

Using 3 as the base and the equation as exponents:

(x-2)/(x+2) = 3^2

(x-2)/(x+2) = 9

x-2 = 9(x+2)

x-2 = 9x + 18

-2 = 8x + 18

-20 = 8x

-20/8 = x

-10/4 = x

-5/2 = x

check:

log_3( -5/2 - 2) - log_3(-5/2+2) =

log_3(-9/2) - log_3(-1/2)

log of negative number not allowed...

the answer does not check

no solution

I also graphed f(x) = log(x-2)-log(x+2)

If you do, it will not cross the line y=2

in fact the x-axis the asymptote