are you sure that you typed that correctly?

{e}^{2t} - {243}^{t} + 143 = 0

is hard

{e}^{2t} - {243}e^{t} + 143 = 0

can be solved using the subsititution you suggested.

x^2 - 243x + 143 = 0

... (quadratic formula)...

x = 243.587 (you will need more decimals when you do it)

or

x = -0.58 (ignore this negative solution. Can you see why?)

Finally:

since x = e^t

243.57 = e^t

ln(243.57) = t