This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Problem 3: Given the equation a^2 + b^2 = 7ab, we need to prove that ((a+b)/(3)) = (1)/(2)( a + b). Step 1: Start with the given equation and add 2ab to both sides to form a perfect square. a^2 + b^2 = 7ab a^2 + b^2 + 2ab = 7ab + 2ab (a+b)^2 = 9ab Step 2: Take the square root of both sides. sqrt((a+b)^2) = sqrt(9ab) a+b = 3sqrt(ab) (Assuming a, b > 0 for the logarithms to be defined and a+b > 0) Step 3: Divide both sides by 3. (a+b)/(3) = sqrt(ab) Step 4: Take the logarithm of both sides. ((a+b)/(3)) = (sqrt(ab)) Step 5: Apply the logarithm property (x^n) = n x. ((a+b)/(3)) = ((ab)^1/2) ((a+b)/(3)) = (1)/(2) (ab) Step 6: Apply the logarithm property (xy) = x + y. ((a+b)/(3)) = (1)/(2) ( a + b) This completes the proof. Proven: ((a+b)/(3)) = (1)/(2)( a + b) Problem 4: Given x = _a bc, y = _b ca, and z = _c ab, we need to prove that xyz = x+y+z+2. Step 1: Use the change of base formula _p q = ( q)/( p). Let's use a common base (e.g., natural logarithm or base 10 logarithm) for . Let a = L_a, b = L_b, c = L_c. x = _a (bc) = ((bc))/( a) = ( b + c)/( a) = (L_b + L_c)/(L_a) y = _b (ca) = ((ca))/( b) = ( c + a)/( b) = (L_c + L_a)/(L_b) z = _c (ab) = ((ab))/( c) = ( a + b)/( c) = (L_a + L_b)/(L_c) Step 2: Calculate the product xyz. xyz = ((L_b + L_c)/(L_a)) ((L_c + L_a)/(L_b)) ((L_a + L_b)/(L_c)) xyz = ((L_b + L_c)(L_c + L_a)(L_a + L_b))/(L_a L_b L_c) Expand the numerator: (L_b + L_c)(L_c + L_a)(L_a + L_b) = (L_b L_c + L_b L_a + L_c^2 + L_c L_a)(L_a + L_b) = L_a L_b L_c + L_a^2 L_b + L_a L_c^2 + L_a^2 L_c + L_b^2 L_c + L_a L_b^2 + L_b L_c^2 + L_a L_b L_c = L_a^2 L_b + L_a^2 L_c + L_b^2 L_a + L_b^2 L_c + L_c^2 L_a + L_c^2 L_b + 2 L_a L_b L_c So, xyz = (L_a^2 L_b + L_a^2 L_c + L_b^2 L_a + L_b^2 L_c + L_c^2 L_a + L_c^2 L_b + 2 L_a L_b L_c)/(L_a L_b L_c) xyz = (L_a^2 L_b)/(L_a L_b L_c) + (L_a^2 L_c)/(L_a L_b L_c) + (L_b^2 L_a)/(L_a L_b L_c) + (L_b^2 L_c)/(L_a L_b L_c) + (L_c^2 L_a)/(L_a L_b L_c) + (L_c^2 L_b)/(L_a L_b L_c) + (2 L_a L_b L_c)/(L_a L_b L_c) xyz = (L_a)/(L_c) + (L_a)/(L_b) + (L_b)/(L_c) + (L_b)/(L_a) + (L_c)/(L_b) + (L_c)/(L_a) + 2 Step 3: Calculate the sum x+y+z. x+y+z = (L_b + L_c)/(L_a) + (L_c + L_a)/(L_b) + (L_a + L_b)/(L_c) x+y+z = ((L_b)/(L_a) + (L_c)/(L_a)) + ((L_c)/(L_b) + (L_a)/(L_b)) + ((L_a)/(L_c) + (L_b)/(L_c)) x+y+z = (L_b)/(L_a) + (L_c)/(L_a) + (L_c)/(L_b) + (L_a)/(L_b) + (L_a)/(L_c) + (L_b)/(L_c) Step 4: Compare xyz and x+y+z+2. From Step 2: xyz = (L_b)/(L_a) + (L_c)/(L_a) + (L_c)/(L_b) + (L_a)/(L_b) + (L_a)/(L_c) + (L_b)/(L_c) + 2 From Step 3: x+y+z = (L_b)/(L_a) + (L_c)/(L_a) + (L_c)/(L_b) + (L_a)/(L_b) + (L_a)/(L_c) + L ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._