SOLUTION: Please help me solve this question Prove the identity ; log N base a . Log N base b + log N base b . Log N base c + log N base c . Log N base a = log N base a . Log N base b . Log

Algebra ->  Test -> SOLUTION: Please help me solve this question Prove the identity ; log N base a . Log N base b + log N base b . Log N base c + log N base c . Log N base a = log N base a . Log N base b . Log      Log On


   

Question 1034314: Please help me solve this question
Prove the identity ; log N base a . Log N base b + log N base b . Log N base c + log N base c . Log N base a = log N base a . Log N base b . Log N base c/ log N base abc
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
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Please help me solve this question
Prove the identity ; log N base a . Log N base b + log N base b . Log N base c + log N base c . Log N base a = log N base a . Log N base b . Log N base c/ log N base abc
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Check that the left side is equal to

1%2F%28log%28N%2C%28a%29%29%2Alog%28N%2C%28b%29%29%29 + 1%2F%28log%28N%2C%28b%29%29%2Alog%28N%2C%28c%29%29%29 + 1%2F%28log%28N%2C%28a%29%29%2Alog%28N%2C%28c%29%29%29.


For it, use the identity log%28x%2C%28N%29%29 = 1%2Flog%28N%2C%28x%29%29, which is universal for logarithms.


Write this sum with the common denominator, which is

log%28N%2C%28a%29%29%2Alog%28N%2C%28b%29%29%2Alog%28N%2C%28c%29%29.

Then the numerator will be 

log%28N%2C%28a%29%29%2Blog%28N%2C%28b%29%29%2Blog%28N%2C%28c%29%29 = log%28N%2C%28abc%29%29 = 1%2Flog%28%28abc%29%2CN%29%29.

Exactly as the right side is.


Proved.