if x^4+y^4=83x^2y^2, then prove that log(x^2-y^2/9)=logx+logy

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We are given the equation:

x⁴ + y⁴ = 83x²y²

Step 1: Express in terms of squares
Rearrange the given equation:

x⁴ + y⁴ - 83x²y² = 0

Using the identity:

(x² - y²)² = x⁴ + y⁴ - 2x²y²

Rewrite the equation as:

(x² - y²)² + 2x²y² = 83x²y²

Simplify:

(x² - y²)² = 81x²y²

Step 2: Take the square root
(x² - y²) = ±9xy

Taking only the positive value:

x² - y² = 9xy

Step 3: Apply logarithm
We need to prove:

log((x² - y²)/9) = log x + log y

Substituting x² - y² = 9xy:

log((9xy)/9) = log x + log y

log(xy) = log x + log y

Since log(xy) = log x + log y (by logarithm property), the given equation is proved.

Thus, the given statement is true.