It is given that log x = 6 and log y = -2. Evaluate (a) log (xy) (b) log (1/sqrt(x))

The questions you've sent are from past ZIMSEC papers (e.g., J2020, N2019, J2018), not the 2024 paper. I will solve the visible questions for you. Question 1: Given x = 6 and y = -2. a) Evaluate (xy). Step 1: Use the logarithm property (AB) = A + B. (xy) = x + y Step 2: Substitute the given values. (xy) = 6 + (-2) = 4 The value is 4. b) Evaluate ((1)/(sqrt(x))). Step 1: Rewrite the expression using exponent rules. ((1)/(sqrt(x))) = (x^-1/2) Step 2: Use the logarithm property (A^B) = B A. (x^-1/2) = -(1)/(2) x Step 3: Substitute the given value for x. -(1)/(2) x = -(1)/(2) × 6 = -3 The value is -3. Question 2: Evaluate: a) 16 ÷ 4. Step 1: Rewrite 16 as a power of 4. ( 16)/( 4) = ((4^2))/( 4) Step 2: Use the logarithm property (A^B) = B A. (2 4)/( 4) Step 3: Simplify the expression. (2 4)/( 4) = 2 The value is 2. b) 5 - 0.5. Step 1: Use the logarithm property A - B = ((A)/(B)). 5 - 0.5 = ((5)/(0.5)) Step 2: Simplify the fraction inside the logarithm. ((5)/(0.5)) = (10) Step 3: Evaluate 10 (assuming base 10). (10) = 1 The value is 1. Question 3: Given 6 = 0.7781 and 5 = 0.6990. a) Calculate 30. Step 1: Rewrite 30 as a product of 5 and 6. 30 = (5 × 6) Step 2: Use the logarithm property (AB) = A + B. (5 × 6) = 5 + 6 Step 3: Substitute the given values and calculate. 5 + 6 = 0.6990 + 0.7781 = 1.4771 The value is 1.4771. b) Calculate 1\,200\,000. Step 1: Express 1\,200\,000 in terms of 10 and a number that can be related to 5 and 6. 1\,200\,000 = (12 × 10^5) Step 2: Use the logarithm property (AB) = A + B. (12 × 10^5) = 12 + 10^5 Step 3: Calculate 12 using the given values. Rewrite 12 = (6 × 10)/(5). 12 = ((6 × 10)/(5)) = 6 + 10 - 5 12 = 0.7781 + 1 - 0.6990 = 1.7781 - 0.6990 = 1.0791 Step 4: Calculate 10^5. 10^5 = 5 10 = 5 × 1 = 5 Step 5: Add the results from Step 3 and Step 4. 1\,200\,000 = 1.0791 + 5 = 6.0791 The value is 6.0791. 3 done, 2 left today. You're making progress.