Is Entropy Extensive in an Ideal Solid?

[cstang52]

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Homework Statement

Starting from S(E,N)=c(N)+3Nk[1+LN(\frac{E}{3Nh\nu})], derive a version of the Entropy, S(E,N) of an ideal solid that is extensive, that is, for which S(\lambdaE,\lambdaN)=\lambdaS(E,N)

Homework Equations

The Attempt at a Solution

Basically have to prove that S(\lambdaE,\lambdaN)=\lambdaS(E,N).

I can set it up, but I don't know how to eliminate terms to get to a form I can work with.

 

[cstang52]

I have it setup like this:

S(λ E,λ N)=λ S(E,N)

c(λN)+3(λN)k[1+ln\frac{λE}{3(λN)h\nu}]=λ{c(N)+3Nk[1+ln\frac{E}{3Nh\nu}]


But 1. I don't know how to reduce the left side, and
2. when I distribute λ through the right side, is it on everything ending up looking like this: c(λN)+λ(3Nk)[λ+ln\frac{λE}{λ(3Nh\nu)}]? Or something else...

 

[haruspex]

ending up looking like this: c(λN)+λ(3Nk)[λ+ln\frac{λE}{λ(3Nh\nu)}]

There's one too many lambdas in there:
c(λN)+λ(3Nk)[1+ln\frac{λE}{λ(3Nh\nu)}]