Any analytical solutions for this?

[languid violet]

so there is something interesting about phi, that being that phi^2 = phi+1

but what we're taking the logarithm of is 2phi+1

rather than just phi +1

so it's phi^2 + phi

so you could decompose the logarithm like this

log_phi(phi(phi+1)) = log_phi(phi)+log_phi(phi+1)

From there I think you can figure the rest out