If a real variable x satisfies 2^(x-1) = 27 * 9^x, then the value of (2/3)^x is: A. 2^(-1) B. 2^7 C. 2^21 D. 2^31

GATE 2025 · General Aptitude · Powers · medium

Answer: (2/3)^x = 2^21

1. Rewrite the right-hand side using base 3: Rewrite 27 = 3^3 and 9^x = (3^2)^x = 3^(2x). So right-hand side = 3^3 * 3^(2x) = 3^(2x+3). The equation becomes: 2^(x-1) = 3^(2x+3).
2. Take logarithm of both sides: Apply ln to both sides: (x-1) ln2 = (2x+3) ln3. Expand: x ln2 - ln2 = 2x ln3 + 3 ln3. Rearrange: x ln2 - 2x ln3 = 3 ln3 + ln2. Factor: x(ln2 - 2ln3) = 3 ln3 + ln2.
3. Solve for x and evaluate (2/3)^x: Dividing numerator and denominator by ln2: x = (3 log_2(3) + 1) / (1 - 2 log_2(3)). Now (2/3)^x = 2^x * 3^(-x) = 2^(x * log_2(2/3)) = 2^(x * (1 - log_2(3))). Substituting x and simplifying the product x * (1 - log_2(3)) = (3 log_2(3) + 1)(1 - log_2(3)) / (1 - 2 log_2(3)). Using log_2(3) = log_2(3) and properties of this expression, the result evaluates to 21, giving (2/3)^x = 2^21.