If p^{x}=q^{y}=r^{z}, where x, y and z are in GP, then consider the following statements : I. p, q and r are in AP. II. \ln p, \ln q and \ln r are in GP. Which of the statements given above is/are correct?

Explanation

Let p^x = q^y = r^z = k. Then p = k^{1/x}, q = k^{1/y}, r = k^{1/z}. Taking the natural log gives \ln p = \frac{\ln k}{x}, \ln q = \frac{\ln k}{y}, \ln r = \frac{\ln k}{z}. Since x, y, z are in GP, their reciprocals \frac{1}{x}, \frac{1}{y}, \frac{1}{z} are also in GP. Thus \ln p, \ln q, \ln r are in GP.

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