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log23和log34

解:log(2)3×log(3)4 =log(2)3×log(2)4/log(2)3 =log(2)4 =log(2)2² =2log(2)2 =2

应用对数换底公式得到哦 log2 3•log3 4•log4 5•log5 6•log6 7•log7 8 =(lg3/lg2)×(lg4/lg3)×(lg5/lg4)×(lg6/lg5)×(lg7/lg6)×(lg8/lg7) =log2 8 =log2 2³ =3

=lg4/lg3 ×lg8/lg3 ×lg3/lg2 ×lg9/lg2 =2lg2/lg3 ×3lg2/lg3 ×lg3/lg2 ×2lg3/lg2 =12

(log23+lOg427)(log34+log98) =(lg3/lg2+lg27/lg4)(lg4/lg3+lg8/lg9) =(lg3/lg2+3lg3/2lg2)(2lg2/lg3+3lg2/2lg3) =(5lg3/2lg2)(7lg2/2lg3) =(5/2)(7/2) =35/4

log23?log34?log45?…?log10231024=lg3lg2?lg4lg3?lg5lg4…lg1024lg1023=lg1024lg2=log21024log2210=10,故选C.

∵an=logn+1(n+2)=lg(n+2)lg(n+1).∴a1?a2?…?ak=lg3lg2?lg4lg3?…?lg(k+2)lg(k+1)=lg(k+2)lg2,∵a1?a2?…?ak=2013,∴lg(k+2)lg2=2013,∴k+2=22013,解得和谐数k=22013-2.故答案为:k=22013-2.

(1)原式=5×(-4)×(-65)×x?23+1?13×y12?12+16=24×1×6y=246y;(2)原式=(lg3lg2+2lg33lg2)(2lg2lg3+3lg22lg3+lg2lg3)+(lg2)2+(2lg2+lg5)×lg5=2+32+1+43+1+23+(lg2)2+(1+lg2)(1-lg2)=6+32+1=172.

(1)lg2?lg50+lg5?lg20-log34?log23?lg2?lg5=lg2(lg5+1)+lg5(lg2+1)-log24log23log23lg2lg5=lg2lg5+lg2+lg5lg2+lg5-2lg2lg5=lg2+lg5=1(2)log2512=log512log525=log56+12log542=a+12b2=2a+b4.

(1)原式=log34×68=log33=1.(2)原式=1+3×[(23)3]23=1+43=73.

(1)(2764)?13+(214)12+12log26=(6427)13+(94)12+12log26=43+32+16=3(2)(log23+log89)?(log34+log98+log32)=(log23+23log23)?(2log32+ 32log32+log32)=53log23?92log32=152

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