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Question
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
Solution
log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2
log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2 `"log"_5^5` (∴ `"log"_5^5` = 1 )
∴ log5 `((x^4 + "y"^4)/(x^4 - "y"^4)) = "log"_5^(5^2)`
∴ `(x^4 + "y"^4)/(x^4 -"y"^4)` = 52 (∴ log a = log b ⇒ a = b)
∴ x4 +y4 = 25(x4 - y4)
∴ x4 + y4 = 25x4 – 25y4
∴ y4 + 25y4 = 25x4 - x4
∴ 26y4 = 24x4
Differentiating w. r. t. x, we get
∴ `26xx4y^3("dy")/("d"x) = 24xx4x^3`
∴ `("dy")/("d"x) = (24xx4x^3)/(26xx4"y"^3)`
∴ `("dy")/("d"x) = (12x^3)/(13"y"^3)`
Hence proved.
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