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

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

Feb 23, 2018

#1
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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

Feb 23, 2018
#2
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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

Feb 23, 2018
#3
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Wondering if  a/b =1 might be considered an extraneous (invalid) solution, as it would result in a

LOG (negative number)   ??

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

Feb 23, 2018
#5
+18049
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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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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?

Feb 23, 2018
#7
+18049
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The chicken !   No wait...... The egg!   No, wait.....

Thanx for the clarification help !

~EP

ElectricPavlov  Feb 23, 2018
#8
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your very welcome...actually it's foghorn leghorn!!!....great to see a good sense of humour

Feb 23, 2018
#9
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Maybe it's a case of which fell off the log first....the chicken or the egg.....???

Feb 23, 2018