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)) #
