Question 61959
The given expression is: 


log<sub>b</sub>x = log<sub>b</sub>(x – 5) – log<sub>b</sub>(x + 7)


This can be further written as: 


0 = log<sub>b</sub>(x – 5) - log<sub>b</sub>(x + 7) - log<sub>b</sub>x 


This implies: 


log<sub>b</sub>(x - 5) - (log<sub>b</sub>(x + 7)+ log<sub>b</sub>x) = 0


log<sub>b</sub>(x - 5)- log<sub>b</sub>((x + 7)(x)) = 0 


log<sub>b</sub>((x - 5)/x(x + 7)) = 0


by the definition we get: 


{{{b^0}}} = {{{x/(x-5)(x+7)}}}


1 = {{{(x - 5)/x(x+7)}}} 


x(x + 7) = x - 5 


This implies:


{{{x^2 + 7x }}} = x - 5 


{{{x^2 + 6x - 5 = 0 }}}


On solving this, we get:


{{{x^2 - 5x - x - 5 = 0}}}


(x - 5) (x - 1) = 0


Thus, x = 5, 1