Consecutive logging

Algebra Level 2

Compute log ⁡ 3 4 ⋅ log ⁡ 4 5 ⋅ log ⁡ 5 6 ⋅ . . . ⋅ log ⁡ 728 729 \log_{3} {4}\cdot \log_{4} {5}\cdot \log_{5} {6}\cdot... \cdot \log_{728} {729}


The answer is 6.

Danish Ahmed by Danish Ahmed , 24, India

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1 solution

Danish Ahmed
Jul 7, 2015

log ⁡ b a = log ⁡ a log ⁡ b \log_{b}a = \dfrac{\log a}{\log b}

The expression =

log ⁡ 4 log ⁡ 3 log ⁡ 5 log ⁡ 4 . . . . . log ⁡ 729 log ⁡ 728 \dfrac{\log 4}{\log 3}\dfrac{\log 5}{\log 4} ..... \dfrac{\log 729}{\log 728}

log ⁡ 729 log ⁡ 3 = log ⁡ 3 6 log ⁡ 3 = 6 \dfrac{\log 729}{\log 3}=\dfrac{\log 3^6}{\log 3} = 6

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