Problem: The product, ab⋅ba\log _{a} b \cdot \log _{b} alogab⋅logba is equal to:
Answer Choices:
A. 111
B. aaa
C. bbb
D. aba bab
E. none of these Solution:
For permissible values of aaa and b,ab⋅ba≡1b, \log _{a} b \cdot \log _{b} a \equiv 1b,logab⋅logba≡1;
or
Let x=abx=\log _{a} bx=logab and y=ba.∴ax=by=\log _{b} a . \quad \therefore a^{x}=by=logba.∴ax=b and by=ab^{y}=aby=a.
∴axy=by=a;∴xy=1;∴ab⋅ba=1\therefore a^{x y}=b^{y}=a ; \quad \therefore x y=1 ; \quad \therefore \log _{a} b \cdot \log _{b} a=1∴axy=by=a;∴xy=1;∴logab⋅logba=1.