The rate of change of the concentration $C$ of a chemical reagent in a reaction is modeled by the differential equation

$$\frac{dC}{dt}=\frac{8t\ln t}{C},t>0$$

where $t$ is the time in seconds. Given that the concentration is 6 units when $t=1$, solve the differential equation to find an expression for the concentration at any time $t$, giving your answer in the form $C^2=f(t)$.