SOLUTION: If a,b,c are distinct positive numbers each different from 1 such that {(log a to the base b)(log a to the base c) – log a to the base a} + {(log b to the base a)(log b to th
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Question 67014: If a,b,c are distinct positive numbers each different from 1 such that {(log a to the base b)(log a to the base c) – log a to the base a} + {(log b to the base a)(log b to the base c) – log b to the b} + {(log c to the base a)(log c to the base b) – log c to the base c} = 0 , then prove that abc = 1
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
If a,b,c are distinct positive numbers each different from 1 such that {(log a to the base b)(log a to the base c) – log a to the base a} + {(log b to the base a)(log b to the base c) – log b to the b} + {(log c to the base a)(log c to the base b) – log c to the base c} = 0 , then prove that abc = 1
USE FORMULA LOG(X) TO BASE Y = LOG(X)/LOG(Y) TO A COMMON BASE
[LOG(A)/LOG(B)][LOG(A)/LOG(C)]-[LOG(A)/LOG(A)]+[LOG(B)/LOG(A)][LOG(B)/LOG(C)]
-[LOG(B)/LOG(B)]+[LOG(C)/LOG(A)][LOG(C)/LOG(B)]-[LOG(C)/LOG(C)]=0
[LOG(A)/LOG(B)][LOG(A)/LOG(C)]+[LOG(B)/LOG(A)][LOG(B)/LOG(C)]+[LOG(C)/LOG(A)][LOG(C)/LOG(B)]
=1+1+1=3
MULTIPLY WITH LOG(A)LOG(B)LOG(C)..THROUGH OUT
LOG^3(A)+LOG^3(B)+LOG^3(C)=3LOG(A)LOG(B)LOG(C)
THE PROBLEM IS NOT TYPED PROPERLY..IT SHOULD BE
{(log a to the base b)(log a to the base c)/[log a*LOGa]...ETC...I THINK
THEN YOU WILL GET
LOG(A)+LOG(B)+LOG(C)=0
LOG(ABC)=LOG(1)
ABC=1
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