If x^4 + y^4 = 14x^2 y^2, then consider the following: I. \log_{10}(x^2 + y^2) = \log_{10} x + \log_{10} y + 2\log_{10} 2 II. \log_{10}(x^2 - y^2) = \log_{10} x + \log_{10} y + \log_{10}2 + 0.5\log_{10} 3 Which of the above is/are correct?
- A. I only
- B. II only
- C. Both I and II ✓
- D. Neither I nor II
Correct Answer: C. Both I and II
Explanation
Adding 2x^2y^2 to both sides gives (x^2+y^2)^2 = 16x^2y^2, meaning x^2+y^2 = 4xy. Taking the log gives \log(x^2+y^2) = \log 4 + \log x + \log y = 2\log 2 + \log x + \log y, so I is correct. Subtracting 2x^2y^2 gives (x^2-y^2)^2 = 12x^2y^2, meaning x^2-y^2 = 2\sqrt{3}xy. Taking the log gives \log 2 + \frac{1}{2}\log 3 + \log x + \log y, so II is also correct.
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