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?
Analyzing the Algebraic Condition
We are given the equation: \(x^4 + y^4 = 14x^2 y^2\) To simplify and relate terms like \(x^2+y^2\) and \(x^2-y^2\), let's consider the squares of these expressions.
Simplifying \(x^2 + y^2\)
Consider the square of \((x^2 + y^2)\): \((x^2 + y^2)^2 = (x^2)^2 + (y^2)^2 + 2x^2 y^2 = x^4 + y^4 + 2x^2 y^2\) Now, substitute the given condition \(x^4 + y^4 = 14x^2 y^2\) into this equation: \((x^2 + y^2)^2 = (14x^2 y^2) + 2x^2 y^2 = 16x^2 y^2\) Taking the square root of both sides (assuming \(x^2+y^2\) is positive, which holds true if \(x\) or \(y\) is non-zero): \(x^2 + y^2 = \sqrt{16x^2 y^2} = 4|xy|\) In the context of logarithmic functions, we usually assume variables are positive. If \(x > 0\) and \(y > 0\), then \(|xy| = xy\). Thus, \(x^2 + y^2 = 4xy\)
Simplifying \(x^2 - y^2\)
Next, consider the square of \((x^2 - y^2)\): \((x^2 - y^2)^2 = (x^2)^2 + (y^2)^2 - 2x^2 y^2 = x^4 + y^4 - 2x^2 y^2\) Substitute the given condition \(x^4 + y^4 = 14x^2 y^2\): \((x^2 - y^2)^2 = (14x^2 y^2) - 2x^2 y^2 = 12x^2 y^2\) Taking the square root of both sides (assuming \(x^2 > y^2\) for the logarithm to be defined): \(x^2 - y^2 = \sqrt{12x^2 y^2} = \sqrt{12}|xy|\) Since \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\), we have: \(x^2 - y^2 = 2\sqrt{3}|xy|\) Assuming \(x > 0\) and \(y > 0\), then \(|xy| = xy\). Thus, \(x^2 - y^2 = 2\sqrt{3}xy\)
Verifying Statement I
Statement I claims: \(\log_{10}(x^2 + y^2) = \log_{10} x + \log_{10} y + 2\log_{10} 2\).
Let's examine the Left Hand Side (LHS) using our derived expression \(x^2 + y^2 = 4xy\): LHS = \(\log_{10}(x^2 + y^2) = \log_{10}(4xy)\) Using the logarithm property \(\log(abc) = \log a + \log b + \log c\): LHS = \(\log_{10} 4 + \log_{10} x + \log_{10} y\) Since \(4 = 2^2\), we can rewrite \(\log_{10} 4\) as \(\log_{10}(2^2) = 2\log_{10} 2\). So, the LHS becomes: LHS = \(2\log_{10} 2 + \log_{10} x + \log_{10} y\) This matches the Right Hand Side (RHS) of Statement I, which is \(\log_{10} x + \log_{10} y + 2\log_{10} 2\). Therefore, Statement I is correct.
Verifying Statement II
Statement II claims: \(\log_{10}(x^2-y^2)=\log_{10} x + \log_{10} y + \log_{10}2+0.5\log_{10} 3\).
Let's examine the Left Hand Side (LHS) using our derived expression \(x^2 - y^2 = 2\sqrt{3}xy\): LHS = \(\log_{10}(x^2 - y^2) = \log_{10}(2\sqrt{3}xy)\) Using the logarithm property \(\log(abc) = \log a + \log b + \log c\): LHS = \(\log_{10} 2 + \log_{10} \sqrt{3} + \log_{10} x + \log_{10} y\) Since \(\sqrt{3} = 3^{1/2}\), we know that \(\log_{10} \sqrt{3} = \log_{10}(3^{1/2}) = 0.5\log_{10} 3\). Substituting this back into the LHS expression: LHS = \(\log_{10} 2 + 0.5\log_{10} 3 + \log_{10} x + \log_{10} y\) This matches the Right Hand Side (RHS) of Statement II, which is \(\log_{10} x + \log_{10} y + \log_{10}2+0.5\log_{10} 3\). Therefore, Statement II is correct.
Conclusion
Based on the analysis, both Statement I and Statement II are derived correctly from the given equation \(x^4 + y^4 = 14x^2 y^2\) using properties of algebra and logarithms.
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