Given equation is #y=logx-5#
Now, for the given equation to make sense, the #log# should give us a proper value. To ensure that it does we should take note that the function doesn't give values for #AAx<=0#. This means for values #x>0# the #log# function gives us proper values.
Since only the #log# function is used in this particular equation to have the variable #x#, it's sufficient to ensure that the value of #x# suit the #log# function.
This means the domain of the function is #0 < x < oo#
Now to find the range.
The #log# function gives us values from the largest of negative to the largest of positive numbers.
If the value input into the function tends to 0, the log function tends to an extremely large negative value (which we'll term as #-oo#). Similarly, if one were to input a very large positive number, the log function gives us a very large positive value (which we'll term as #oo#)
So the range of the function is #-oo < x < oo#
Now, one would be curious as to why only the #log# function is being considered here and why not the #-5# term. Well that's because the #-5# term doesn't affect the domain or range at all.
Why not the domain? Well, the domain is considered as the set of values for which #x# in particular accepts. Since the value #-5# has no hand here as it is out of the function (or any function except #f(x)# for that matter) we ignore it.
Why not the range? The range is considered as the set of value for which an input of #x# a value is received. The term #-5# doesn't influence this because the defined numbers #oo# and #-oo# are called "infinities" which aren't influenced in any way by well-defined numbers.
If you don't get it, don't fret to just ask.
