On an logarithmic equation by primes

Let $[\cdot ]$ be the floor function. In this paper we show that every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], \end{equation*} where $p_1,p_2,p_3$ are prime numbers. We also establish an asymptotic formula for the number of such representations, when $p_1,p_2,p_3$ do not exceed given sufficiently large positive number.