Log_(2)(2x + 1) - 5 log_(4)( {x}^(2) ) + 4 log_(2)(x)

Question

`log_(2)(2x + 1) - 5 log_(4)( {x}^(2) ) + 4 log_(2)(x)`

Answer 1

Evaluate each expression individually and then combine like terms:

5 log₄(x²) = 5
`(log (x^(2)) )/(log (4) )` = 5
`(log (x^(2)) )/(log (2^(2)) )` = 5
`(2 log (x) )/(2 log (2) )` = 5 log₂x = log₂x⁵

4 log₂(x) = log₂(x)⁴ = log₂(x⁴)

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log₂(2x + 1) - 5 log₄(x²) + 4 log₂(x)

= log₂(2x + 1) - log₂x⁵ + log₂(x⁴)

= log₂[(2x + 1)*(x⁴) ÷ x⁵]

= log₂
`((2x+1)x^(4) )/(x^(5))`

= log₂
`(2x+1 )/(x)`