Question

Prove that `("log"_"p" x)/("log"_"pq" x)` = 1 + log_p q

Answer 1

L.H.S.
= `("log"_"p" x)/("log"_"pq" x)`

= `((("log" x)/("log""p")))/((("log"x)/("log""pq"))`

= `("log"x)/("log""p") xx ("log""pq")/("log"x)`

= `("log""pq")/("log""p")`

= `("log""p" + "log""q")/("log""p")`

= `1 + ("log""q")/("log""p")`
= 1 + log_p q
= R.H.S.
Hence proved.