How do you express this as a single logarithm

log base(2) x + log base(4) x + log base(8)x - log base(16) x

...use the change of base formula so that each term is [base 2] logarithm:

log [base 4] x = (log [base 2] x) / (log [base 2] 4) = (log [base 2] x) / 2

similarly...

log [base 8] x = (log [base 2] x) / 3

log [base 16] x = (log [base 2] x) / 4

can you simplify now ? I hope so...try, get involved...

id est

{(19/12) log [base 2] x = log [base 2] (x^(19/12) }

log base(a) b = Ln(b)/Ln(a)

=(Ln(x)/Ln(2)) + (Ln(x)/2Ln(2)) + (Ln(x)/3Ln(2)) + (Ln(x)/4Ln(2))

= ((25/3)Ln(x))/(4Ln(2))
= Ln(x^(25/12))/Ln(2)

= log base(2) (x^(25/12))

make wooden planks out of the logs.