What the purpose of a speech is? Ideally, no one likes to trick another! Thus, people should ideally try to convey their ideas. This transfer of reasoning is logos: to persuade by logic. Here is my (rhetorical) question: what else is logical? Is "persuading" by touching emotions something really (logically) valid? Of course the fact that an advertisement has nice visual and sonic effects does not mean that the product itself is nice. So, if the purpose of an argument is not to trick the audience but to report the truth and speak objectively, which ideally must be, then logos is the only way that one can persuade another AND be a critical thinker. So, pathos is not an ideal way of persuading people (but if your purpose is not to find the truth but to sell your products then pathos is important for you to make money. It does not mean that your product is really good or not and you do not care. The only thing you want is money.)
Ethos, is a bit different. This is a matter of probability: if person X is authoritative in scientific community, it can be inferred that he has had lots of great (and true) ideas, so the (conditional) probability that person X statement about Y is true this time is high. (of course higher than a highschool student) So, ethos does NOT prove that person X is right this time. It just means that the probability of it is high (but not 1).
As conclusion, if the purpose is proving, the logos is the only way that someone has. Others are not sufficiently logical to prove something. (of course a speaker cannot be his own statement's proof and our emotions - mind, more generally - is not necessarily a representation of reality.)