(log以4为底3+log以8为底3)×(log以3为底5+log以9为底5)×(log以5为底2+log以25为底2)求值
题目
(log以4为底3+log以8为底3)×(log以3为底5+log以9为底5)×(log以5为底2+log以25为底2)求值
答案
工具1 log(a^n)(M)=(1/n)log(a)(M)(n∈R)
工具2 log(a)(N)=log(b)(N)/log(b)(a)= lnN/lna=lgN/lga
工具2‘ log(a)b×log(b)c×log(c)a=1
(log以4为底3+log以8为底3)×(log以3为底5+log以9为底5)×(log以5为底2+log以25为底2)
=(1/2log以2为底3+1/3log以2为底3)×(log以3为底5+1/2log以3为底5)×(log以5为底2+1/2log以5为底2)
=(5/6log以2为底3)×(3/2log以3为底5)×(3/2log以5为底2)
=5ln3/6ln2*3ln5/2ln3*3ln2/2ln5=15/8
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