Bound on the Kolmogorov complexity of integers

Question

I am reading Elements of Information Theory (Thomas M. Cover, Joy A. Thomas, 2nd edition) in which the following theorems are given (page 475--476):

1. For any integer $n$

$$ K(n) \leq \log^* n + c. $$

2. There are an infinite number of integers such that $K(n) > \log n$.

Where $K(n)$ is the Kolmogorov complexity of $n$ and $\log^* n$ is the iterated logarithm of $n$.

Assuming both statements true, this implis that there are an infinite number of integers $n$ such that

$$ \log n < \log^* n + c. $$

Knowing the growing rate of $\log$ and $\log^*$ this does not seem possible if $c$ is independent of $n$... Is it? If not, which one of the two above statements is false?