I don't understand how can I reach the correct alternative ("None of the above is valid for all a,b>0), is there anybody whose understood could explain me?
(a) $\frac{1}{2}[\ln(3)-\log_a(b)]$
(b) $\sqrt{\ln(3)-\log_a(b)}$
(c) $\frac{1}{2}[3-\frac{1}{\log_b(a)}]$
(d) $\sqrt{3+\log_b(a)}$
(e) None of the above is valid for all a,b>0.
I've reached the alternative (c) :
$\log_a(\sqrt{a^3})-\log_a(\sqrt{b})=\log_a(a^{\frac{3}{2}})-\log_a(b^{\frac{1}{2}})=\frac{1}{2}(3-\log_a(b))=\frac{1}{2}(3-\frac{1}{\log_b(a)})$