Unexpected result from FullSimplify?

Greetings,
I’m trying to simplify an expression by using FullSimplify[x-y], where x is the original expression and y is the version I step-by-step reform to a simpler version. As long as the result is zero, I’m doing well. The expression with which I am working will not simplify to zero. If I simplify it slightly, it will. Examples are in the attached notebook, as rendering in text makes for difficult reading.

Thank you for any help that you can provide.

Best regards,
Neil Geismar

Perhaps the problem is that the expression

E^(\[Alpha] ((2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha])))

is replaced with

(-\[Delta] \[Lambda]T)^((2 \[Alpha])/(-\[Alpha] + \[Rho]))

before Simplify has a chance to work, and this expression is a little too complicated. A workaround is ti Inactivate the Log, so that the replacement does not take place:

Inactivate[
  E^(-\[Alpha] (t + 4 T - (
      2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha]))) -
   E^(-\[Alpha] (t + 4 T)) *
    E^(\[Alpha] ((2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha]))),
  Log] // Simplify