Logarithm

y = log ₂ (2x)

y= log ₄ (16 + x)

Change of base theorem

log ( 2x) / log 2  =  log (16 + x) / log 4

log (2x) / log 2  = log (16 + x) / log 2^2

log (2x) / log 2  =  log (16 + x) / [ 2 log 2]            {multiply both sides by log 2 }

log (2x)  =  log (16 + x) / 2

2 log (2x)  = log (16 + x)

log (2x)^2  =  log (16 + x)     implies that

2x^2  = 16 + x

4x^2  - x  - 16  =  0        { x must be positive }

x = [ 1 + sqrt [ 1 + 256 ] ]  / 8  =   [ 1 + sqrt (257) ] /  8