有关对数
已知log4(底数) 27=a,log5(底数) 2=b,求lg2(真数)及lg3(真数)
(1): log(5)2=b, 1/log(2)5=b, log(2)5=1/b, log(2)5+1=1/b+1, log(2)5+log(2)2=(b+1)/b, log(2)10=(b+1)/b, lg2=1/log(2)10=b/(b+1). (2): log(4)27=a, log(2^2)3^3=a, (3/2)*log(2)3=a, log(2)3=2a/3, lg3/lg2=2a/3,[log(a)b=log(c)b/log(c)a] lg3=lg2*2a/3=2ab/3(b+1)
解:∵log5 2=b ∴lg2/lg5=b ∵lg5=1-lg2 ∴lg2/(1-lg2)=b ∴lg2=b/(b+1) ∵log4 27=a ∴(3/2)*log2 3=a,log2 3=2a/3 ∵log2 3=lg3/lg2=2a/3,lg2=b/(b+1) ∴lg3/[b/(b+1)]=2a/3 ∴lg3=(2a/3)*[b/(b+1)]=2ab/(3b+3) 综上所述,lg2=b/(b+1),lg3=2ab/(3b+3)
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