Mathematica, \$\approx\$ 7.67 score
25 23 bytes, arbitrary precision, final score \$ \frac{23}{3}\$
N[#/Log[#]Log@#,PrimePi[#]+1]&PrimePi@#+1]&
-2 bytes, courtesy of @ExpiredData
Here is a benchmarking solution. The TIO link has funny-looking output but the function works just fine when ran locally with Mathematica's softwareTry it online!.