Mathematica gets the parametric integral wrong:
Assuming[0 < a < 1, Integrate[Log[1 - a/x], {x, 1, \[Infinity]}]]
but it gets it right when the parameter a is a number:
With[{a = 1/2}, Integrate[Log[1 - a/x], {x, 1, \[Infinity]}]]
If you force Evaluate on the integral, the wrong result will win the evaluation race, before the parameter a gets a numeric value. Remove Evaluate and the parameter a will become numeric before the integral is evaluated:
Manipulate[
Integrate[Log[1 - a/x], {x, 1, [Infinity]}], {a, 0.001, 0.999}]