Anish Singhal
Last Activity: 6 Years ago
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.
