SOLUTION: Can someone please explain why logbx/ logby does not equal logbx - logby?

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Question 274779: Can someone please explain why logbx/ logby does not equal logbx - logby?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To prove something false, we usually resort to a counterexample. A counterexample is an explicit example in which proves a theorem or equation false (usually by a contradiction of some sort).

So let's say that x=b and y=b. This would then mean that


Now plug x=b and y=b into log%28b%2C%28x%29%29-log%28b%2C%28y%29%29 to get log%28b%2C%28x%29%29-log%28b%2C%28y%29%29=log%28b%2C%28b%29%29-log%28b%2C%28b%29%29=1-1=0


So in short, log%28b%2C%28x%29%29%2Flog%28b%2C%28y%29%29=1 and log%28b%2C%28x%29%29-log%28b%2C%28y%29%29=0 when x=b and y=b.


Clearly 1%3C%3E0 which means that log%28b%2C%28x%29%29%2Flog%28b%2C%28y%29%29%3C%3Elog%28b%2C%28x%29%29-log%28b%2C%28y%29%29 when x=b and y=b.

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Here's another way of looking at it.


By the change of base formula, log%28b%2C%28x%29%29%2Flog%28b%2C%28y%29%29=log%28y%2C%28x%29%29

By another identity, log%28b%2C%28x%29%29-log%28b%2C%28y%29%29=log%28b%2C%28x%2Fy%29%29


So let's assume that log%28b%2C%28x%29%29%2Flog%28b%2C%28y%29%29=log%28b%2C%28x%29%29-log%28b%2C%28y%29%29. If this is the case, then log%28y%2C%28x%29%29=log%28b%2C%28x%2Fy%29%29


Equate the bases and arguments to get y=b and x=x%2Fy. The second equation simplifies to 1=1%2Fb. Solve for b to get b=1.


Now if b=1, this means that which is impossible (you can't divide by zero).