No, it has not taken off exponentially yet in the way the Tony Seba video demonstrates with a picture of 5th Ave in NYC in the year 1900 where there are lots of horses used for transportation, and only one car, compared with a picture of the same street in 1913, where the street is full of cars, and there is only one horse to be found.
There's the formal definition of "exponential", and then there is "exponential" in the sense of completely disruptive and transformative as in the example Tony Seba gives. I am referring to the latter.
One of the comments there points it out: The government keeps passing subsidies for solar that nominally expire, then renewing them again before they expire. Then the government's models assume the subsidies expire when the law says they will rather than getting renewed time and again.
tl;dr The International Energy Agency kept predicting, year after year, in the face of all the evidence, that solar production would plateau.