Solve log4(x+4) – 2 log4(x+1) =1/2 for x.  

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Solve `log_4(x+4)-2log_4(x+1)=1/2` .

First, simplify using logarithm rules:

`log_4(x+4)-log_4(x+1)^2=1/2`
`log_4((x+4)/(x+1)^2)=1/2`

Now exponeniate both sides with base 4:

`(x+4)/(x+1)^2=4^(1/2)` noting that `4^(1/2)=2` we cross-multiply:

`x+4=2(x+1)^2`
`x+4=2x^2+4x+2`
`2x^2+3x-2=0`
`(2x-1)(x+2)=0`

So `x=1/2` or `x=-2` . But -2 is not in the domain of the original function.

Thus the solution is `x=1/2` .

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