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Algebra Difficulty 5.1 AIME, harder Find the answer

2. Find the value of log⁡223+log⁡234+log⁡245+log⁡256+log⁡267+log⁡278log⁡33⋅log⁡43⋅log⁡54⋅log⁡85⋅log⁡76⋅log⁡87\frac{\log _{2} \frac{2}{3}+\log _{2} \frac{3}{4}+\log _{2} \frac{4}{5}+\log _{2} \frac{5}{6}+\log _{2} \frac{6}{7}+\log _{2} \frac{7}{8}}{\log _{3} 3 \cdot \log _{4} 3 \cdot \log _{5} 4 \cdot \log _{8} 5 \cdot \log _{7} 6 \cdot \log _{8} 7}.

A number or a short expression. Spacing and $ signs are ignored.

Solution

2. -6 .
 Original expression =log⁡2(23⋅34⋅45⋅56⋅67⋅78)lg⁡2lg⁡3⋅lg⁡3lg⁡4⋅lg⁡4lg⁡5⋅lg⁡5lg⁡6⋅lg⁡6lg⁡7⋅lg⁡7lg⁡8=log⁡228lg⁡2lg⁡8=log⁡22−2lg⁡23lg⁡2=−213=−6. \begin{array}{l} \text { Original expression }=\frac{\log _{2}\left(\frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdot \frac{5}{6} \cdot \frac{6}{7} \cdot \frac{7}{8}\right)}{\frac{\lg 2}{\lg 3} \cdot \frac{\lg 3}{\lg 4} \cdot \frac{\lg 4}{\lg 5} \cdot \frac{\lg 5}{\lg 6} \cdot \frac{\lg 6}{\lg 7} \cdot \frac{\lg 7}{\lg 8}} \\ =\frac{\log _{2} \frac{2}{8}}{\frac{\lg 2}{\lg 8}}=\frac{\log _{2} 2^{-2}}{\frac{\lg 2}{3 \lg 2}}=\frac{-2}{\frac{1}{3}}=-6 . \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.