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 :)

I do not see how the "If ..." part is related to the demanded computation. (Also, $a = 1 + \log_{72}2$.) — Torsten Schoeneberg, Jan 11 '20 at 03:24
I don't think it can be represented in terms of $a$ as $7, 11, 13$ are prime numbers. Because of this, you won't be able to split it into those bases. — Sam, Jan 11 '20 at 03:44

score 0 · Answer 1 · answered Jan 11 '20 at 05:04

0

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.

answered Jan 11 '20 at 05:04

Ross Millikan

374,822

In the given question by using the formula above we can conclude that $log_{1001} ^{501}= alog_{1001} ^{72} log_{144} ^{501}$. Ie., $log_{1001} ^{501}=0.77431*a$. – Ch.Siva Ram Kishore Jan 11 '20 at 05:59
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@Ch.SivaRamKishore: Yes, but usually when questions are asked like that the variable assigned is useful, giving a simpler form of the answer. Here there seems no relationship between $a$ and the value we are to evaluate. – Ross Millikan Jan 11 '20 at 06:38

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

1 Answers1