Suppose I have a case like
T(n)=2T(n/4)+log(n). a=2, b=4, f(n)=log(n)
That should be case 1 because
n^(1/2)>log(n). There is also a lambda in case 1.
f(n)=O(n^((1/2)-lambda). Is this correct? And how can I find this lambda?
The constant lambda is important: its purpose is to avoid considering the bizarre cases that lie between Case 1 and Case 2. Since big-O is an upper bound only and not a lower bound, smaller choices of lambda are "better" in the sense that they cover more functions. Since lambda must be positive, however, there is no "best" choice of lambda. Lambda = 10^-3 should get you through enough examples to see why most treatments of the Master Theorem do not make a production out of choosing lambda.
Epsilon can be 1/4 since
So by case 1 of the master theorem T(n)=Θ(nlogba)=Θ(n(1/2)).