Can provide us a linear convex combination of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$ a better approximation for $\pi(x)$, the prime counting function? Or not, is better $Li(x)$ when we consider the ratio $\frac{\pi(x)}{Li(x)}$ for large real numbers $x$? I know that $Li(x)$ is better than $\frac{x}{\log x}$.
Let a fixed $0<\theta<1$, for such linear convex combination $L_\theta(x)$ we can compute using Prime Number Theorem $$\lim_{x\to\infty}\frac{L_\theta(x)}{\pi(x)}=1,$$ but it does improve the approximation that provide us the logarithmic integral $Li(x)$?
Question. Let $\theta$ a real number such that $0<\theta<1$, does improve $$\theta\int_2^x\frac{1}{\log(t)}dt+(1-\theta)\frac{x}{\log(x)}$$ the aproximation of Prime Number Theorem? I say the asymptotic $\sim 1$ for $\pi(x)$ given by $Li(x)$ or $x/\log x$. Thanks in advance.