Problem: The number of real values of x satisfying the equality (loga​x)(logb​x)=loga​b, where a>0,b>0,aî€ =1,bî€ =1, is:
Answer Choices:
A. 0
B. 1
C. 2
D. a finite integer greater than 2
E. not finite
Solution:
Let loga​x=y∴x=ay∴logb​x=logb​ay=ylogb​a
∴{(loga​x)(logb​x)=loga​b} implies {y⋅ylogb​a=loga​b}
Since logb​a=1/loga​b,y2=(loga​b)2,y=loga​b or y=−loga​b
∴loga​x=loga​b or loga​x=loga​b−1∴x=b or x=b−1=1/b
or
Let logb​x=y∴x=by∴(loga​by)(logb​by)=loga​b
(yloga​b)(ylogb​b)=loga​b,y2loga​b=loga​b∴y2=1∴y=1 or y=−1∴x=b or x=b^
(logb​x)(loga​x)=loga​x(logb​x)=loga​b∴xlogb​x=b
∴(logb​x)(logb​x)=logb​b=1∴logb​x=1 or logb​x=−1∴x=b or x=b^