I'm using the following rules:
Rule 1: log(x*y) = log(x)+log(y)
Rule 2: log(x/y) = log(x)-log(y)
Rule 3: log(x^y) = y*log(x)
Rule 4: sqrt(x) = x^(1/2)
log[ (x^2*y)/(sqrt(z)) ] = log[ x^2*y ] - log[ sqrt(z) ] .... Use Rule 2
log[ (x^2*y)/(sqrt(z)) ] = log[ x^2 ] + log[ y ] - log[ sqrt(z) ] .... Use Rule 1
log[ (x^2*y)/(sqrt(z)) ] = log[ x^2 ] + log[ y ] - log[ z^(1/2) ] .... Use Rule 4
log[ (x^2*y)/(sqrt(z)) ] = 2*log[ x ] + log[ y ] - (1/2)*log[ z ] .... Use Rule 3
log[ (x^2*y)/(sqrt(z)) ] = 2*a + b - (1/2)*c .... Use Substitution
So
log[ (x^2*y)/(sqrt(z)) ]
will break down into
2*log[ x ] + log[ y ] - (1/2)*log[ z ]
which is equivalent to
2a + b - (1/2)c
when a = log(x), b = log(y), c = log(z)
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So that's why the answer is choice (4) 2a + b - (1/2)c
