That depends on the slope of the line and how far we are above the 0 utility threshold for a life worth living. I found this book with a table (table B21 on p. 264) of estimated historical GDP going back to the year 0, and there have been less than six doublings in that time. Current countries with per capita GDP at the same levels as pre-1900 Western Europe (and even year 0 Western Europe) are included in some of those analyses that found the log-linear fit, and their self-reported well-being is in line with the regression that fits the rest of the world. The log-linear relationship might break down if we go far enough into the past (or future), to places that are poor enough (or rich enough), or to societies that are different enough so that GDP won't be a good measure of their material quality of life, but the data I've seen suggest that the relationship is more robust than I would've expected.
The Stevenson & Wolfers paper that I linked uses self-report measures of welfar including happiness, satisfaction with life, and amount of smiling, and finds similar log-linear relationships with income on all of them (though with different slopes and intercepts), which suggests that this relationship will apply to whichever definition of utility we use.
GDP measures essentially how good we are at making widgets - and while widgets are useful, it is a very weak and indirect measure of welfare. For example UK GDP per capita doubled between 1975 and 2007 - and people's quality of life indeed improved - but it would be extremely difficult to argue that this improvement was "doubling", and that the gap between 2007's and 1975's quality of life is greater than between 1975's and hunter-gatherer times.
It's not essential to this post, but my very quick theory is that we overestimate GDP thanks to economic equivalent of Amdahl's Law - if someone's optimal consumption mix consisted of 9 units of widgets and 1 unit of personalized services - and their purchasing power increased so now they can acquire 100x as many widgets, but still the same number of services as before - amount of the mix they can purchase increased only 9x, not 90x you'd get by weighted average of original consumption levels (and they spend 92% of their purchasing power on services now). The least scalable factor - whichever it is - will be the bottleneck.
If we're unhappy with GDP there are alternative measures like HDI, but they're highly artificial. It would be very easy to construct completely different measures which would "feel" about as right.
Fortunately there exists a very natural measure of welfare, which I haven't seen used before in this context - preference utilitarian lotteries. Would you rather live in 1700, or take a 50% chance of living in 2010 or 700? Make a list of such bets, assign numbers coherent with bet values (with 100 for highest and 0 for your lowest value) and you're done! By averaging many people's estimates we can hopefully reduce the noise, and get some pretty reasonable welfare estimates.
And now disclaimer time. This approach has countless problems, here are just a few but I'm sure you can think about more.
I tried to think about such series of bets and my results are:
This seems far more reasonable than GDP's illusion of exponentially accelerating progress.
I used this Ruby code to convert bets to values on scale of 0 to 100 (bets ordered by preference, not chronologically):