I am trying to lower bound the following function, $n \ge 3$ is a natural number:

$$l(n):=\frac{\log(n)}{\log(n)-\frac{1}{n}(\tau(n)\log(\tau(n))+(n-\tau(n))\log(n-\tau(n))}$$

where $\tau(n)$ counts the number of divisors. Is this function exponential or polynomial in terms of $\log(n)$?

Thanks for your help.

I am trying to lower bound the following function, $n \ge 3$ is a natural number:

$$l(n):=\frac{\log(n)}{\log(n)-\frac{1}{n}(\tau(n)\log(\tau(n))+(n-\tau(n))\log(n-\tau(n)))}$$

where $\tau(n)$ counts the number of divisors. Is this function exponential or polynomial in terms of $\log(n)$?

Thanks for your help.