Evaluate log8 log16

Here are the solutions to the questions: Question 8: In the figure below, O is the centre of the circle. If ABC = 30^, find AOC. Step 1: Identify the relationship between the angle at the center and the angle at the circumference. The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference. Here, arc AC subtends AOC at the center and ABC at the circumference. Step 2: Apply the circle theorem. AOC = 2 × ABC Step 3: Substitute the given value. Given ABC = 30^. AOC = 2 × 30^ Step 4: Calculate the value of AOC. AOC = 60^ This corresponds to option D. The final answer is D. 60^. Question 9: Evaluate _8 16. Assumption: The question "Evaluate log8 log16" is interpreted as _8 16, as this is a common format for such problems in multiple-choice questions and leads to one of the options. Step 1: Set the expression equal to x. x = _8 16 Step 2: Convert the logarithmic equation to an exponential equation. 8^x = 16 Step 3: Express both sides of the equation with the same base. The common base for 8 and 16 is 2. (2^3)^x = 2^4 2^3x = 2^4 Step 4: Equate the exponents. 3x = 4 Step 5: Solve for x. x = (4)/(3) Upon reviewing the options, (4)/(3) is not present. Let's re-evaluate the interpretation. If the question intended to be _16 8, let's check that. Re-evaluation based on options: If the question was _16 8: Step 1: Set the expression equal to x. x = _16 8 Step 2: Convert to an exponential equation. 16^x = 8 Step 3: Express both sides with base 2. (2^4)^x = 2^3 2^4x = 2^3 Step 4: Equate exponents. 4x = 3 Step 5: Solve for x. x = (3)/(4) This value matches option D. Given the options, it is highly probable that the question intended to be _16 8. The final answer is D. (3)/(4). Question 10: If y is a whole number such that 2y + 1 = 4 7, find the least value of y. Step 1: Subtract 1 from both sides of the congruence. 2y + 1 - 1 4 - 1 7 2y 3 7 Step 2: Find the multiplicative inverse of 2 modulo 7. We need a number a such that 2a 1 7. By testing values: 2 × 1 = 2 7 2 × 2 = 4 7 2 × 3 = 6 7 2 × 4 = 8 1 7 So, the inverse of 2 7 is 4. Step 3: Multiply both sides of the congruence by the inverse of 2, which is 4. 4 × 2y 4 × 3 7 8y 12 7 Step 4: Simplify the congruence. Since 8 1 7 and 12 5 7: 1y 5 7 y 5 7 Step 5: Determine the least whole number value for y. The congruence y 5 7 means y can be 5, 12, 19, . The least whole number value is 5. This corresponds to option A. The final answer is A. 5. Last free one today — make it count tomorrow, or type /upgrade for unlimited.