finite fields - Discrete logarithm - strange polynomials

If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number $\log_{\omega} x$ between $1$ and $p-1$ such that $\omega^{\log_{\omega} x} = x$.

$\log_{\omega} x$ will be identified with its remainder class in $\mathbb{Z}/p\mathbb{Z}$.

I was interested in finding the interpolating polynomial in $\mathbb{Z}/p\mathbb{Z}[x]$ of minimal degree for this $\log_{\omega}$ function (for fun?) and looked at a few examples where $\omega = 3$ can be taken as a fixed primitive root. Some examples:

modulo $5$, $\log_3(x) = 4x^3 + 3x^2 + 2x$.

modulo $7$, $\log_3(x) = 5x^5 + 2x^4 + 4x^3 + 6x^2 + 3x$.

modulo $17$, $\log_3(x) =$

$$10x^{15} + 16x^{14} + 3x^{13} + 11x^{12} +14x^{11} + 12x^{10} + 13x^9 + 9x^8 + 5x^7 + 6x^6 + 4x^5 + 7x^4 + 15x^3 + 2x^2 + 8x.$$

I notice that the coefficients in each case are exactly the numbers $2$ to $p-1$ up to permutation. Is there something behind this? It seems highly improbable if it's a coincidence.