If X2 + Y2 = 7xy, Prove that Log ( X − Y 3 ) = 1 2 (Log X + Log Y) - Mathematics

If x² + y² = 7xy, prove that `"log"((x - y)/3) = (1)/(2)` (log x + log y)

Sum

x² + y² = 7xy
⇒ x² + y²- 2xy = 7xy - 2xy
⇒ (x + y)² = 9xy
⇒ `((x + y)/3)^2` = xy

⇒ `((x + y)/3) = sqrt(xy)`
Considering log both sides, we get
`"log"((x + y)/3) = "log"(xy)^(1/2)`

⇒ `"log"((x + y)/3) = (1)/(2)"log"(xy)`

⇒ `"log"((x + y)/3) = (1)/(2)["log" x + "log" y]`.

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Expansion of Expressions with the Help of Laws of Logarithm

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Chapter 10: Logarithms - Exercise 10.2

Chapter 10 Logarithms
Exercise 10.2 | Q 26

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