If 5^{a_{1}} = 6 , 6^{a_{2}} = 7 , 7^{a_{3}} = 8 \left(..........624\right)^{a_{620}} = 625 , then find \prod_{r = 1}^{620} a

Engineering

Mathematics

base change theorem

Question

If $5^{a_{1}} = 6, 6^{a_{2}} = 7, 7^{a_{3}} = 8 {..........624}^{a_{620}} = 625,$ then find $\prod_{r = 1}^{620} a_{r}$ .

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a₁ = log₅ 6 , a₂ log₆ 7, …… a₆₂₀ = log₆₂₄ 625
$∴ \prod_{r = 1}^{620} \log_{5} 6$ · log₆ 7 …… log₆₂₄ 625 = log₅ 625 = 4