logarithm problem - four tuple How many distinct four

Javion Kerr

Javion Kerr

Answered question

2022-03-22

logarithm problem - four tuple
How many distinct four tuple (a,b,c,d) of rational numbers are there with
alog102+blog103+clog105+dlog107=2005
Can we proceed like this :
Using loga+logb=log(ab) and mloga=logam
log102a3b5c7d=2005
Please guide how to proceed further..

Answer & Explanation

Ireland Vaughan

Ireland Vaughan

Beginner2022-03-23Added 14 answers

You have
2a3b5c7d=102005=2200552005.
So this works if a=c=2005 and b=d=0.
To see if other solutions exist, observe that you'd get
2a20055c2005=3b7d.
All the exponents are rational, so we have
2n1n25n3n4=3n5n67n7n8.
Raising both sides to the power n2n4n6n8, we get all integer exponents. By uniqueness of ' factorizations, that can happen only if all of the integer exponenents are 0. Therefore there can be no other solutions.
Adan Berry

Adan Berry

Beginner2022-03-24Added 12 answers

Hint: 102005=2a3b5c7d

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