+0  
 
0
111
1
avatar

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

 Feb 21, 2020
 #1
avatar+111438 
+1

2log ( a -2b)  =  log a  + log b

 

By log properties, we can write

 

log (a - 2b)^2  = log (ab)

 

This implies that

 

(a -2b)^2   = ab       simplify

 

a^2 - 4ab  + 4b^2   = ab     subtract  ab  from both sides

 

a^2  -5ab +  4b^2    = 0        factor as

 

(a - 4b) ( a - b)  =   0

 

So...either

 

a - 4b   = 0                        or                  a  - b  = 0

 

a  = 4b  ⇒ a/b  = 4                                    a  =  b                  

                               

 

The second answer (in red)  would  mean that  b would have to be negative in log (a - 2b)......but this isn't defined  for log  b

 

So

 

a/b  = 4     

 

 

cool cool cool

 Feb 21, 2020

1 Online Users