<p><strong>27.</strong> If \(x, y, z\) are in G.P. and \(a^x = b^y = c^z\), then</p>
<p>\(\log_b a = \log_c b\)</p>
<p>\(\log_b b = \log_c c\)</p>
<p>\(\log_a a = \log_b b\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: If x, y, z are in G.P., then y² = xz. Combined with a^x = b^y = c^z = k (say), we can express a, b, c in terms of k and use the G.P. condition to find the relationship between logarithms.
<p><strong>Step 1:</strong> Let a^x = b^y = c^z = k (some constant)</p><p><strong>Step 2:</strong> Taking logarithms: x·ln(a) = y·ln(b) = z·ln(c) = ln(k)</p><p>Therefore: ln(a) = ln(k)/x, ln(b) = ln(k)/y, ln(c) = ln(k)/z</p><p><strong>Step 3:</strong> Since x, y, z are in G.P.: y² = xz</p><p><strong>Step 4:</strong> Now observe that 1/x, 1/y, 1/z form a sequence where:</p><p>(1/y)² = 1/y² and (1/x)·(1/z) = 1/(xz) = 1/y²</p><p>Therefore: (1/x)·(1/z) = (1/y)²</p><p><strong>Step 5:</strong> This means ln(a), ln(b), ln(c) are in G.P., which means a, b, c are in G.P.</p><p>∴ ln(a), ln(b), ln(c) are in G.P. (or equivalently: a, b, c are in G.P.)</p>
Correct Answer: A