Minimization with constraint involving logs in Julia

I want to solve \(\frac{1}{N} \sum_{i=1}^N x_i^2\) subject to \(\frac{1}{N} \sum_{i=1}^N \log x_i \geq 0\).

In the above, N is the size of the vector x whose \(i^{th}\) element is \(x_i\). I think the solution should be \(x_j = 1\), for all \(j=1,\ldots, N\). How do I code this in Julia? I appreciate that I could replace the constraint involving a sum by a product but in the true application I have in mind (which involves logs) this may not be possible. Basically I want to minimize a function which is convex and straightforward but with a constraint involving logs. I tried the following but it doesn't work. In the below, ncol is the size of n (which is denoted x above)

model = Model(Gurobi.Optimizer)
@variable(model, 1000.0 >= n[1:ncol] >= 0.001)
@objective(model, Min, sum(n[i]^2 for i in eachindex(n)) / ncol)
@expression(model, expr, sum(log(n[i]) for i in 1:ncol))
@NLconstraint(model, expr >= 0.0)
optimize!(model)

Can anyone tell me how I could incorporate a constraint on the log of x? The code is Julia.

Thank you.

Minimization with constraint involving logs in Julia

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