The paper studies (logarithmic) real valuations of the polynomial ring \(\mu:K[T]\longrightarrow \mathbb R\) that are proper valuations when restricted to the field \(K\). Crucial notions are the Apéry base \(\{\omega_i(\mu)\}_{i\geq 0}\), where \(\omega_0(\mu)=\mu(T)\) and, given an integer \(i\geq 1\), \[ \omega_i(\mu)=\sup\{\mu(f) : f\in K[T] \text{ monic, with deg}(f)=i\} \] and the iterated sequence \(\{(\mu_i, d_i, \gamma_i)\}_{i=0}^t\) of valuations associated with \(\mu\), which is defined inductively, starting with \(\mu_0\in \mathrm{Val}(K[T])\), \(\mu_0\left(\sum_{0\leq i\leq d}a_iT^i\right)=\text{ min }\{\mu(a_i)+i\mu(T), 0\leq i\leq d\}\), \(d_0=1\) and \(\gamma_0=\mu(T)\). After establishing a number of relevant properties, a relation between these central notions is given in Theorem 21: Under certain assumptions, \(\omega_i(\mu)=\sum_{j=0}^{k-1}s_j\gamma_j\), where the sum \(=\infty\) if \(k-1=t\) and \(\gamma_t=\infty\). In particular \(\omega_{d_j}(\mu)=\gamma_j\), \(1\leq j\leq t\). Moreover \(\omega_i(\mu)\leq (i/d_{k-1})\gamma_{k-1}\) and the equality holds if and only if \(d_{k-1}\) divides \(i\). One of the results is a factorization Theorem 29 for monic members \(f\in K[T]\) of given degree and \(\mu(f)\) the Apéry base of \(\mu\) which establishes strong properties of their irreducible factors. One special case of this result are some earlier results by M. Merle, R. Ephraim and A. Granja for algebroid plane curve singularity. Proposition 31 justifies why the sequence \(\{\gamma_i/d_i\}_{i=1}^t\) may be called the polar invariants of the valuation \(\mu\).