But during the analysis i tried to see what power of x equals the behaviour of log function up to a constant.
But interestingly it is dominated by all the inverse monomials (polynomial with one power) near zero and the usual monomials at infinity. If you want to see compare log(x) and x^m for all m as x approaches to zero and for infinity as second case.
Final line: Near zero , log is similar to an infinite stretch
Near infinity, it is similar to a constant function.