log base(2) x + log base(4) x + log base(8)x - log base(16) x
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...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 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) }
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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))
=(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))
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make wooden planks out of the logs.