Relative error definition of entropy

[kraigandrews]

Homework Statement

Problem-5: In the class we saw that property of additivity of the entropy for systems in thermalcontact relies on the approximation that we compute entropy of the most probable state ln(g1g2)maxinstead of full ln g(N,U) = ln(P,U1)
g(N1,U1)g(N2,U −U1)). The error made turns out to be really
small provided that each systems contains large number of particles. For the example of spins with
N1 = N2 = N/2 >> 1 we found
ln g(N, s) = ln(g1g2)max +1/2ln(N/8).
(4)
Consider now a special case when one system is large having N1 = 10E22 spins but the other is relatively
small with only N2 = 10 spins. Take s = 0 and compute the relative error in the definition
of entropy. Is it still a good idea to use such definition of entropy when N1 >> N2? [1 pt]

Homework Equations

The Attempt at a Solution

Ok so I honestly haven't a clue where to start I am just looking for type of guidance. My first idea would be to find g1 and g2 then from there I am totally lost.

 

[Nate810]

I know that g(N,U) = ln(P,U1) g(N1,U1)g(N2,U −U1)). But I don't know how to use it or really what it means.