I am not sure if this is the right forum, perhaps it should be in the basic algebra forum but it is the sort of question that involves concepts rather then techniques.
To find a solution set for:
(1): ln(2x+4) = x^2
must one use a graphing method?
Either the intersection of y1 = ln(2x+4) and y2 = (x^2),
the intersection of y = ln(2x+4) -x^2 with the x-axis (the zeros of the function)?
If one must use a graphing technique, why is that, what is missing or needed, some new identity?
I note too that in applied mathematics there is such a thing as getting a rough check of your work using "dimensional analysis" but in mathematics, dimensions are not a factor. I can see that this is the case because all expressions made over, say the real numbers, must ultimately reduce to a real number therefore you are always dealing with a correspondence between pure numbers, attached "dimensions" then a separate issue. Still, in this case it does seem that the dividing line between a log equation that is solvable, at least by the basic algebra techniques that I know of, and one that is not, is delineated by mixed dimensions as it were, namely terms that represent exponents (i.e. ln(2x+4) ), with those that do not (i.e. x^2).
In addition to the above question I suppose what I am also sort of grasping for is how one can look at variations of the above equation and know that a graphing solution is the only way to go.