This doesn't really belong in "Linear and Abstract Algebra", it is just a matter of basic algebra.
From $\displaystyle \theta= \theta_0\left(1- e^{\frac{t}{T}}\right)$ you can divide both sides by $\displaystyle \theta_0$ to get $\displaystyle \frac{\theta}{\theta_0}= 1- e^{\frac{t}{T}}$. Now add $\displaystyle e^{\frac{t}{T}}$ to both sides and subtract $\displaystyle \frac{\theta}{\theta_0}$ from both sides to get $\displaystyle e^{\frac{t}{T}}= 1- \frac{\theta}{\theta_0}$
Take the natural logarithm of both sides to get
$\displaystyle \frac{t}{T}= ln\left(1- \frac{\theta}{\theta_0}\right)$
Can you finish it from there?