Question

Find x and y, if `("log"x)/("log"5) = ("log"36)/("log"6) = ("log"64)/("log"y)`

Answer 1

`("log"x)/("log"5) = ("log"36)/("log"6) = ("log"64)/("log"y)`
Considering the first equality
`("log"x)/("log"5) = ("log"36)/("log"6)`

⇒ `("log"x)/("log"5) = ("log"6^2)/("log"6) = (2"log"6)/("log6)` = 2
⇒ log x = 2log5 = log 5² = log25
∴ x = 25
Considering the second equality
`("log"36)/("log"6) = ("log"64)/("log"y)`

⇒ `("log"6^2)/("log"6) = (2"log"6)/("log"6) = 2 = ("log"8)/("log"y)`

⇒ log y = `("log"64)/(2) = ("log"8^2)/(2) = (2"log"8)/(2)` = log8
∴ y = 8.