No, best way to visulaize testosteroine levels is to look at it like a gaussian shape curve (probably not accurate but it helps me visulaize it) and not exponential curve like you are saying. During puberty, the testosterone levels are high, that why you get secondary sexual charac (caused by adrogens), etc. But as a person get older, the amount of testosterone declines, only enough to maintain the characteristics.
Hope this helps!