Geometric Progression With Logarithms

This question links progressions through the logarithm's transformation of products into sums, a connection JEE Advanced uses to convert multiplicative structure into additive structure. Since x, y, z are in GP, the middle term satisfies y^2 = xz. Taking logarithms of both sides gives 2 log y = log x + log z, which is precisely the defining condition for log x, log y, log z to be in arithmetic progression. Thus the logarithms form an AP. Option geometric progression is wrong because logarithms convert the constant ratio into a constant difference, not a ratio. Option harmonic progression has no basis here, since reciprocals of logarithms are not forced to be arithmetic. Option no fixed progression contradicts the clean derived condition. Hence the answer is arithmetic progression. This duality runs both ways: applying the exponential function, the inverse of the logarithm, turns an arithmetic progression of exponents back into a geometric progression of values, so logarithms and exponentials act as a translation dictionary between additive and multiplicative structure, a theme JEE Advanced returns to repeatedly. Plausibility check: for the GP 1, 10, 100 in base ten the logarithms are 0, 1, 2, which are equally spaced and therefore arithmetic, directly confirming the general result.