Here,
log(x^2+4)-log(x+2)=2+log(x-2)
=>log(x^2+4)-log(x+2)-log(x-2)=2
=>log(x^2+4)-{log(x+2)+log(x-2)}=2
Using : logM+logN=log(MN)
=>log(x^2+4)-log[(x+2)(x-2)]=2
=>log(x^2+4)-log(x^2-4)=2
Using : logM-logN=log(M/N)
log((x^2+4)/(x^2-4))=2
(i)If it is common logarithm-logarithm to base 10 ,then
log_10 ((x^2+4)/(x^2-4))=2
:.(x^2+4)/(x^2-4)=10^2,where,x^2!=4=>color(red)(x!=+-2
x^2+4=100(x^2-4)
:.100x^2-400-x^2-4=0
99x^2-404=0
:.x^2=404/99~~4.08
:.x=+-sqrt4.08~~2.02
But, x~~-2.02 will make log(x-2) meaningless.
:. x~~2.02
Note that most of the textbooks use logx as,
logarithm to base 10
(ii)If it is color(blue)"natural logarithm ??-logarithm to base e
" ,then
log_e ((x^2+4)/(x^2-4))=2
:.(x^2+4)/(x^2-4)=e^2,where,x^2!=4=>color(red)(x!=+-2
x^2+4=e^2(x^2-4)
:.e^2x^2-4e^2-x^2-4=0
:.e^2x^2-x^2=4e^2+4
:.x^2(e^2-1)=4(e^2+1)
:.x^2=4((e^2+1)/(e^2-1))
:.x=2sqrt((e^2+1)/(e^2-1))
