What is (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36 ?

The solution to (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36 is 9(ln(2)-ln(3))+163/36-(ln(2/3))/3

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simplify (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36

simplify

\frac{- ln ( \frac{2}{3} ) - \frac{1}{3}}{3} + 9 ln (\frac{2}{3}) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

9 (ln (2) - ln (3)) + \frac{163}{36} - \frac{ln ( \frac{2}{3} )}{3}

Decimal

1.01374 \dots

Solution steps

\frac{- ln ( \frac{2}{3} ) - \frac{1}{3}}{3} + 9 ln (\frac{2}{3}) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{2}{3}) = ln (2) - ln (3)

= \frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} + 9 (ln (2) - ln (3)) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

Group like terms

= \frac{3}{2} + \frac{1}{9} + \frac{1}{36} + \frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} + 9 (ln (2) - ln (3)) + 3

\frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} = \frac{- 1 - 3 ln ( \frac{2}{3} )}{9}

= \frac{3}{2} + \frac{1}{9} + \frac{1}{36} + \frac{- 3 ln ( \frac{2}{3} ) - 1}{9} + 9 (ln (2) - ln (3)) + 3

Combine the fractions

\frac{1}{9} + \frac{- 3 ln ( \frac{2}{3} ) - 1}{9} : - \frac{ln ( \frac{2}{3} )}{3}

= \frac{3}{2} - \frac{ln ( \frac{2}{3} )}{3} + \frac{1}{36} + 9 (ln (2) - ln (3)) + 3

Combine the fractions

3 + \frac{3}{2} + \frac{1}{36} : \frac{163}{36}

= 9 (ln (2) - ln (3)) + \frac{163}{36} - \frac{ln ( \frac{2}{3} )}{3}

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What is (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36 ?
The solution to (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36 is 9(ln(2)-ln(3))+163/36-(ln(2/3))/3