For the following problem i am stuck with how to rearrange the exponents so that i can find t. I understand if there was only 1 exponent as then i can convert it to log and find t but as there are 2 exp components i am not sure on how to simplify these.
Let us say that you want the zero of $$f(t)=1.71 e^{-26.18 t}-11.7 e^{-3.82 t}+5$$ Since the exponents do not show a ratio of $2$, $3$ or $4$, the equation cannot reduce to a polynomial.
Only numerical methods will solve the problem.
Whet you can notice is that $$f'(t)=44.694 e^{-3.82 t}-44.7678 e^{-26.18 t}$$ cancels for $x\approx 0.0000738$ and the second derivative test shows that this is a minimum. So, you have a unique solution to the equation.
By inspection, $f(0.1)\approx -2.86045$ and $f(0.5)\approx 3.26746$. So, let us start Newton method using $x_0=0.3$. The iterates would be $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.300000 \\ 1 & 0.209721 \\ 2 & 0.222005 \\ 3 & 0.222287 \end{array} \right)$$ which is the solution for six significant figures.