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Let x, y, z be real numbers satisfying the system: log₂(x*y*z - 3 + log₅(x)) = 5 log₃(x*y*z - 3 + log₅(y)) = 4 log₄(x*y*z - 3 + log₅(z)) = 4 Find the value of |log₅(x)| + |log₅(y)| + |log₅(z)|.

1. 225
2. 125
3. 365
4. 265

Correct answer: 265

Solution

Setting up: a+b+c = (35-P)+(84-P)+(259-P) = 378-3P. Also a+b+c = log₅(P). So log₅(P) = 378-3P. Testing P=5^(k): trying P=125 (5³): log₅(125)=3, 378-3*125=378-375=3. Yes! P=125. Then a=35-125=-90, b=84-125=-41, c=259-125=134. |a|+|b|+|c|=90+41+134=265.

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