if x2 +y2=10xy then prove that 2log(x-y) =logx+logy+3log2
Answers
Answer:
No
Step-by-step explanation:
Given 2log(x+y)=logx+logy−2log3 . Show that x2+y2=7xy .
Long-term goals or a short-term goals, aim to meet them all.
ANSWER: NO! 2log (x + y) = log x + log y – 2log 3 does not show that x² + y² = 7xy; It shows instead that x² + y² = –17xy/9.
PROOF:
By the “logarithm of a product” property, we have:
2log(x + y) = log (xy) – 2log 3
Now, by the “logarithm of a power” property, we have:
log (x + y)² = log (xy) – log 3²
log (x + y)² = log (xy) – log 9
Now, by the “logarithm of a quotient” property, we have:
log (x + y)² = log (xy/9)
Now, let n = log (x + y)² and m = log (xy/9), then, substituting, we have the equality:
n = m
By the definition of logarithms and due to the fact that logarithms are exponents, the two logarithmic statements, n = log (x + y)² and m = log (xy/9), can be rewritten into their equivalent exponential forms as follows (We’re assuming that the logarithms that we’re dealing with in this problem are common logarithms (logarithms to the base 10):
10ⁿ = (x + y)² and 10ᵐ = xy/9.
Now, by a property of exponents which says that “if b > 0, b ≠ 1, and “m” and “n” are real numbers, then bⁿ = bᵐ if and only if n = m”; therefore, we have the following equality:
10ⁿ = 10ᵐ
(x + y)² = xy/9
x² + 2xy + y² = xy/9
9(x² + 2xy + y²) = (xy/9)9
9x² + 18xy +9y² = xy
9x² + 9y² = –18xy + xy
9x² + 9y² = –17xy
x² + y² = –17xy/9