If $\log_{72}144=a$. Compute $\log_{1001}501^{2019!}$

Question

Surfing on the web i found a question that i think is bit interesting :0
The problem:
If $\log_{72}144=a$. Compute $\log_{1001}501^{2019!}$

I tried to factorize $72, 144, 1001$ and $501$, but $1001=7*11*13$ and $501=3*167$, and i couldn't figure it out more than this, can someone help me with this nice question?

Thanks in advance for any help :)

Answer 1

It will be enormous. $\log_{1001}501 \approx 0.8998$, so the number you want is about $0.8998 \cdot 2019!$ Alpha says it is about $1.71988 \cdot 10^{5798}$. It will give you lots more digits if you want. I don't know how to use $a$ in this.