Question 128052
{{{log(5,(x+1))+log(5,(x-3))=1}}}


First thing is to use the rule that {{{log(b,a)+log(b,c)=log(b,ac)}}}, so


{{{log(5,(x+1)(x-3))=1}}}


Next, we note that if {{{log(b,a)=y}}} then {{{b^y=x}}}, so


{{{5^1=(x+1)(x-3)}}}


{{{5=x^2-2x-3}}}


{{{x^2-2x-8=0}}}


{{{(x-4)(x+2)=0}}}


{{{x=4}}} or {{{x=-2}}}


However, {{{x=-2}}} means that {{{x-3<0}}} and {{{log(b,x)}}} is not defined for {{{x<0}}}, therefore we need to exclude the second root.


The solution set consists of the single element 4.