I want to find a function $f(n)$ where $$f(n) > c \times \log{n}, \quad c\text{ is arbitrary constant}$$ ,and there is no function $f'(n)$ where $$f(n)>f'(n)>c \times \log{n}$$
I can set $f(n) = \log{n}\times \log^{*}{n}$, which is greater than $c \times \log{n}$. But I think that maybe there is a function $f'(n)$ which is less than $f(n)$.
Can you suggest a function $f(n)$ such that $\log{n}\times \log^{*}{n} > f(n) > c \times \log{n}?$