[7].log的一些題目。20點

以下的 log(A)表示是以10為底的log

7.

(log2(5)+log4(1/5))(log5(2)+log25(1/2))
=[log(5)/log(2)+log2^2(5^-1)]*[log(2)/log(5)+log5^2(2^-1)]
=[log(5)/log(2)+(-1/2)*log(5)/log(2)]*[log(2)/log(5)+(-1/2)*log(2)/log(5)]
=[(1/2)*log(5)/log(2)]*[(1/2)*log(2)/log(5)]
=1/4

8.

log4(28/15)-2log4(3/14)+6log16(6/7)-2log16(2/5)
=[log4(28)-log4(15)]-2[log4(3)-log4(14)]+6[log16(6)-log16(7)]-2[log16(2)-log16(5)]
=[log4(7*4)-log4(3*5)]-2[log4(3)-log4(7*2)]+6[(1/2)log4(2*3)-(1/2)log4(7)]-2[(1/2)log4(2)-(1/2)log4(5)]
=[log4(7)+log4(4)-log4(3)-log4(5)]-[2log4(3)-2log4(7)-2log4(2)]+[3log4(2)+3log4(3)-3log4(7)]-[log4(2)-log4(5)]
=[log4(7)+1-log4(3)-log4(5)]-[2log4(3)-2log4(7)-1]+[3/2+3log4(3)-3log4(7)]-[1/2-log4(5)]
=(1+2-3)log4(7)+(-1-2+3)log4(3)+(-1+1)log4(5)+1+1+3/2-1/2
=3

9.

log4(3)-log9(25)*log5(32)
=log4(3)-[log3^2(5^2)]*log5(2^5)
=log4(3)-2/2*log3(5)*5log5(2)
=log(3)/log(4)-[log(5)/log(3)]*5[log(2)/log(5)]
=(1/2)*log(3)/log(2)-5log(2)/log(3)
=(1/2)log2(3)-5log3(2)

11

log2(log2(log3(81))
=log2(log2(log3(3^4)))
=log2(log2(4))
=log2(2)=1

12

2^3+log4(36)-3log9(25)
=8+log4(4*9)-3log3^2(5^2)
=8+1+log4(9)-3log3(5)
=9+log2(3)-3log3(5)

第9跟12題若題目沒錯,且沒給log(2),log(3),log(5)的值,最後答案應該就是這樣了

7. (log2(5)+log4(1/5))(log5(2)+log25(1/2))

log2(5)+log4(1/5)=log2(5)-0.5log2(5)
=0.5log2(5)=0.5log(5)/log(2)=...(1)
log5(2)+log25(1/2)=log5(2)-0.5log5(2)
=0.5log5(2)=0.5log(2)/log(5)...(2)
(1)×(2)
(log2(5)+log4(1/5))(log5(2)+log25(1/2))=0.25

8. log4(28/15)-2log4(3/14)+6log16(6/7)-2log16(2/5)
全部換成以4為底
log4(28/15)-2log4(3/14)+3log4(6/7)-log4(2/5)
=log4[(28×14^2 ×6^3 ×5)/(15×3^2 ×7^3 ×2)
=log4(64)=3

9. log4(3)-log9(25)×log5(32)
=log(3)/[2log(2)]-5log(2)/log(3)

^^"...若題目為
(1)
[log4(3)-log9(25)]×log5(32)
=5/2×[log(3)/log(5)-log(2)/log(3)]
(2)
log4(3)×log9(25)×log5(32)=2.5

11. log2(log2(log3(81))
log3(81)=4
log2(4)=2
log2(2)=1
∴log2(log2(log3(81))=1

12. 2^3+log4(36)-3log9(25)
=8+log4(4)+log4(9)-6log9(5)
=9+[log(3) /log(2)] -3[log(5) /log(3)]

sorry~我不知道有沒有會錯題目的意思
如果我有會錯意or題目有更新，請指教
我會再修改答案的~^^