Prove that if () = Θ(()), then ln(()) = Θ(ln(()))

Question

Let () and () be asymptotically positive functions, and assume that lim →∞ () = ∞. Prove that if () = Θ(()), then ln(()) = Θ(ln(()))

Answer 1

Hint: given $C_1 $ you need to find $C_2 $ for inequalities $$\begin{array}{} \ln C_1+ \ln g \leqslant C_2 \ln g \\ C_2\ln g \leqslant \ln C_1+ \ln g \end{array}$$