Answer
$$ y= (3^{18} 9\ln 3) \ t+3^{18}(1- 18\ln 3).$$
Work Step by Step
Let $ f(t)=3^{9t} $, then $ f'(t)=3^{9t}9 \ln 3$ and the slope of $ f $ at $ t=2$ is given by
$$ m=f'(2)=3^{18} 9\ln 3.$$
The equation of the tangent line is
$$ y= (3^{18} 9\ln 3) \ t+c.$$
Since the function and the tangent line coincide at $ t=2$, we have
$$ c=3^{18}- (3^{18} 9\ln 3) \ (2)=3^{18}(1- 18\ln 3) .$$ Finally, we get
$$ y= (3^{18} 9\ln 3) \ t+3^{18}(1- 18\ln 3).$$
