As long as its crypto primitives remain secure (SHA-256, Elliptic Curve crypto, RIPEMD), if the Bitcoin economy continues to grow 5x, 10x, 20x, etc, then its value should increase 5x, 10x, 20x, etc, because there is a fixed number of coins available.
Both of these assumptions are likely. So I would say, yes, this exchange rate should last (of course there will be some bumps on the road, bubbles, crashes, etc).
That's exactly how it should grow - in relation to the growth of the economy. But I worry all these posts about the latest Bitcoin price will push it again on the speculation train.
There is slow growth in coins. But if it takes off then we know all of the primitives to make a second bitcoin. Establish one with a price peg to existing bitcoin. If you maintain it long and well enough, it will become accepted as equivalent. Once that happens you've done a big one-time increase in the supply.
In fact I would recommend doing this at some point. Because otherwise bitcoin has built-in deflation. That is bad economically because it discourages investment in anything else, like building businesses.
It's asymptotic growth of coins. There will never be more than x number of bitcoins. (I forget the value x is exactly, but it is a finite, pre-definded number).
So what you would need to do is have an account in both currencies with large amounts of currency in it, and publicly advertise that you will trade across currencies in some predetermined ratio. If your account has enough currency to succeed in handling people who might deliberately try to trade against you, then you will successfully peg currencies.
Admittedly an attempt to provide a peg like this might fail. But what is in it for the entity trying to do it? Quite simply, that entity presumably starts with a much larger amount of the new currency than they used for establishing the peg. If you can successfully create the peg, your net worth decreased by the amount of bitcoin you had to accumulate to create the peg, and increased by the amount of other currency that you had. So you make a handsome profit. (Conversely you have the risk that people will set about mining your new currency and converting it all to bitcoin through your peg, then when you have to abandon the peg you've lost the bitcoin you had to purchase to try to establish said peg.)
There are a number of tricks that can improve your odds of establishing a successful peg. But that's the principle.
Both of these assumptions are likely. So I would say, yes, this exchange rate should last (of course there will be some bumps on the road, bubbles, crashes, etc).