<p>Sum of values of \(x\) and \(y\) satisfying \(\log_x(\log_3(\log_x y)) = 0\) and \(\log_y 27 = 1\) is:</p>
Step-by-Step Solution
Key Concept: We need to solve two logarithmic equations simultaneously. The first equation simplifies using the property that log_a(b) = 0 means b = 1, and the second directly gives us a relationship between y and the base.
<p><strong>Step 1: Simplify the second equation</strong></p><p>From log_y(27) = 1, we have y^1 = 27, so y = 27</p><p><strong>Step 2: Use the first equation with y = 27</strong></p><p>Substitute y = 27 into log_x(log_3(log_x(27))) = 0</p><p><strong>Step 3: Apply the property log_a(b) = 0 ⟹ b = 1</strong></p><p>log_x(log_3(log_x(27))) = 0 means log_3(log_x(27)) = 1</p><p><strong>Step 4: Apply log_a(b) = c ⟹ b = a^c</strong></p><p>log_3(log_x(27)) = 1 means log_x(27) = 3^1 = 3</p><p><strong>Step 5: Solve for x</strong></p><p>log_x(27) = 3 means x^3 = 27</p><p>Therefore x^3 = 27 ⟹ x = 3</p><p><strong>Step 6: Verify the solution</strong></p><p>Check x = 3, y = 27 in the first equation:</p><p>log_3(log_3(log_3(27))) = log_3(log_3(3)) = log_3(1) = 0 ✓</p><p><strong>Step 7: Calculate sum</strong></p><p>x + y = 3 + 27 = 30</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C