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Find a/b when 2log(a-2b) = log a + log b .

waffles  Feb 23, 2018
 #1
avatar
+1

2log(a-2b) = loga + logb  so log(a-2b)2 = logab

 

therefore  (a-2b)2 = ab

 

expanding : a2 -4ab + 4b2 = ab

 

ie              a2  - 5ab + 4b2 = 0

 

divide both sides by b2 : a2/b2 -5ab/b2 + 4 =0

 

ie              (a/b)- 5(a/b) + 4 = 0

 

solving the quadratic for a/b gives (a/b - 1)(a/b - 4) = 0

 

therefore  a/b =1 or a/b = 4

Guest Feb 23, 2018
 #2
avatar+91027 
+1

2 log (a - 2b  )  = loga + log b

 

log (a - 2b)^2  = log (a*b)

 

This implies that

 

(a - 2b)^2  =  ab

 

a^2 - 4ab + 4b^2  = ab

 

a^2 - 5ab + 4b^2  = 0

 

(a - 4b) (a - b)  = 0

 

This implies that

 

a = 4b   ⇒   a/b  =  4      or

 

a - b  = 0  ⇒   a  = b  ⇒   a/b  =  1

 

 

cool cool cool

CPhill  Feb 23, 2018
 #3
avatar+13590 
+1

Wondering if  a/b =1 might be considered an extraneous (invalid) solution, as it would result in a

LOG (negative number)   ??

ElectricPavlov  Feb 23, 2018
 #4
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No I don't think so as squaring a negative gives a positive

 

ie 2log(a-2b) =log (a-2b)2

Guest Feb 23, 2018
 #5
avatar+13590 
+1

That may be true, but on the left you still have a LOG (negative number)    .....So I am confused.

  2 Log (a-2b)   =  log(a-2b)^2    

       seems invalid as the left side of the equation is invalid if (a-2b) is negative  

Does anyone know for SURE?

Help !

 

  ~EP

ElectricPavlov  Feb 23, 2018
 #6
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0

Perhaps the author of the question didn't believe it an issue otherwise they would have included that a-2b >= 0 in the question. I do see your dilemma but if you remember back to when you learnt all about logs and I guess it depends on how your teacher treated the topic. A logarithm as we know is the power of a base to obtain a given number. From this definition we get a whole new mathematics topic part of which is simplifications. The actual log law in question is that if we have a log of a power then :  logbxn  =  nlogbx  not the other way round. Of course the other way has been adopted also to help with simplification. Maybe there are those who think its a case of which came first the chicken or the egg?

Guest Feb 23, 2018
 #7
avatar+13590 
+1

The chicken !   No wait...... The egg!   No, wait.....cheeky

   Thanx for the clarification help !    

~EP

ElectricPavlov  Feb 23, 2018
 #8
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0

your very welcome...actually it's foghorn leghorn!!!....great to see a good sense of humour

Guest Feb 23, 2018
 #9
avatar+91027 
0

Maybe it's a case of which fell off the log first....the chicken or the egg.....???

 

 

cool cool cool

CPhill  Feb 23, 2018

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