Basic properties of Log

Question

If !!(logx)/(b-c)= (logy)/(c-a)= (logz)/(a-b)!!. Then,

(a) !!xyz=1!!

(b) !!x^ay^bz^c=1!!

(c) !!x^(b+c)y^(c+a)z^(a+b)=1!!

(d) All of these

Solution

!!(logx)/(b-c)= (logy)/(c-a)= (logz)/(a-b)=k!!

!!=> logx = k(b-c), logy = k(c-a) and logz = k (a-b)!!

!!=> x = 10 ^(k(b-c)), y = 10 ^(k(c-a)) and z= 10 ^(k(a-b))!!

So for option (a),

!!xyz= [10 ^(k(b-c))][10 ^(k(c-a))][10 ^(k(a-b))]!!

!!=> xyz = 10^0=1!!

Similary put the value of !!x,y,z!! in other two options.

We get option(b) and (c) are also correct.

Therefore, correct option is (d).

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