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(log⁡(x−2)3y)−64−x2−y2\left(\frac{\log(\sqrt{x}-2)}{3y}\right)-\sqrt{64-x^{2}-y^{2}}

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Full solution

Q. (log⁡(x−2)3y)−64−x2−y2\left(\frac{\log(\sqrt{x}-2)}{3y}\right)-\sqrt{64-x^{2}-y^{2}}
  1. Identify Properties: Identify the properties of logarithms and square roots to simplify the given expression (log⁡(x−2))/(3y)−64−x2−y2(\log(\sqrt{x}-2))/(3y)-\sqrt{64-x^2-y^2}. For the logarithm part, we will use the property that allows us to express the logarithm of a power as a multiple of a logarithm. For the square root part, we will simplify the square root expression.
  2. Apply Logarithm Property: Apply the property of logarithms to the term (log⁡(x−2))/(3y)(\log(\sqrt{x}-2))/(3y).\newlineThe property is log⁡b(an)=n⋅log⁡b(a)\log_b(a^n) = n \cdot \log_b(a). Here, we have a square root, which is equivalent to the power of 1/21/2. So we can rewrite the logarithm as:\newline(1/2⋅log⁡(x−2))/(3y)(1/2 \cdot \log(x-2))/(3y).
  3. Simplify Square Root: Simplify the square root expression 64−x2−y2\sqrt{64-x^2-y^2}. We recognize that 64−x2−y264-x^2-y^2 could be a difference of squares if x2+y2x^2 + y^2 equals 6464. However, without additional information, we cannot simplify this further. We leave it as 64−x2−y2\sqrt{64-x^2-y^2}.
  4. Combine Results: Combine the results from Step 22 and Step 33 to write the final simplified expression.\newlineThe final expression is (16y)⋅log⁡(x−2)−64−x2−y2(\frac{1}{6y}) \cdot \log(x-2) - \sqrt{64-x^2-y^2}.

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