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Arithmetic Sequence of Logs

Source: AMC 12 2008A Problem 16

February 17, 2008
logarithmsarithmetic sequenceAMC

Problem Statement

The numbers log⁡(a3b7) \log(a^3b^7), log⁡(a5b12) \log(a^5b^{12}), and log⁡(a8b15) \log(a^8b^{15}) are the first three terms of an arithmetic sequence, and the 12th 12^\text{th} term of the sequence is log⁡bn \log{b^n}. What is n n? <spanclass=′latex−bold′>(A)</span> 40<spanclass=′latex−bold′>(B)</span> 56<spanclass=′latex−bold′>(C)</span> 76<spanclass=′latex−bold′>(D)</span> 112<spanclass=′latex−bold′>(E)</span> 143 <span class='latex-bold'>(A)</span>\ 40 \qquad <span class='latex-bold'>(B)</span>\ 56 \qquad <span class='latex-bold'>(C)</span>\ 76 \qquad <span class='latex-bold'>(D)</span>\ 112 \qquad <span class='latex-bold'>(E)</span>\ 143
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