Since that is a difference of exponentials, taking the logarithm of both sides will not help.
You could start by dividing through by e^{8.875t} to get -.02e^{8.875t}= 0.14- 1.14e^{7.75}.
Now, 7.75 divides into 8.875 1.1452 times. That is if we let x= e[sup]7.75[/tex] that equation becomes x^{1.1452}= 0.14- 1.14 x.
But since that exponent is not an integer, that is not a polynomial and I don't believe there is any non-numerical method to solve either it or the original equation.