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logarithm of logarithm of number

Source: Romanian District Olympiad 2011, Grade X, Problem 4

October 8, 2018
logarithmsalgebraa implies b

Problem Statement

a) Show that , if a,b>1 a,b>1 are two distinct real numbers, then log⁡alog⁡ab>log⁡blog⁡ab. \log_a\log_a b >\log_b\log_a b.
b) Show that if a1>a2>⋯>an>1 a_1>a_2>\cdots >a_n>1 are n≥2 n\ge 2 real numbers, then log⁡a1log⁡a1a2+log⁡a2log⁡a2a3+⋯+log⁡an−1log⁡an−1an+log⁡anlog⁡ana1>0. \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0.
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