What is 5ln(3/4)+5/2+(3(-ln(3)-3ln(3/4)))/2 ?

The solution to 5ln(3/4)+5/2+(3(-ln(3)-3ln(3/4)))/2 is 5ln(3/4)+(5-3ln(81/64))/2

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simplify 5ln(3/4)+5/2+(3(-ln(3)-3ln(3/4)))/2

simplify

5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ln ( 3 ) - 3 ln ( \frac{3}{4} ) )}{2}

5 ln (\frac{3}{4}) + \frac{5 - 3 ln ( \frac{81}{64} )}{2}

Decimal Notation

0.70824 \dots

Solution steps

5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ln ( 3 ) - 3 ln ( \frac{3}{4} ) )}{2}

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

3 ln (\frac{3}{4}) = ln ((\frac{3}{4})^{3})

= 5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ln ( 3 ) - ln ( ( \frac{3}{4} ) ^{3} ) )}{2}

Rewrite

- ln (3) - ln ((\frac{3}{4})^{3})

- (ln (3) + ln ((\frac{3}{4})^{3}))

= 5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ( ln ( 3 ) + ln ( ( \frac{3}{4} ) ^{3} ) ) )}{2}

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (3) + ln ((\frac{3}{4})^{3}) = ln (3 (\frac{3}{4})^{3})

= 5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ln ( 3 ( \frac{3}{4} ) ^{3} ) )}{2}

3 (\frac{3}{4})^{3} = \frac{81}{64}

= 5 ln (\frac{3}{4}) + \frac{5}{2} + \frac{3 ( - ln ( \frac{81}{64} ) )}{2}

Combine the fractions

\frac{5}{2} + \frac{3 ( - ln ( \frac{81}{64} ) )}{2} : \frac{5 - 3 ln ( \frac{81}{64} )}{2}

= 5 ln (\frac{3}{4}) + \frac{- 3 ln ( \frac{81}{64} ) + 5}{2}

= 5 ln (\frac{3}{4}) + \frac{5 - 3 ln ( \frac{81}{64} )}{2}

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What is 5ln(3/4)+5/2+(3(-ln(3)-3ln(3/4)))/2 ?
The solution to 5ln(3/4)+5/2+(3(-ln(3)-3ln(3/4)))/2 is 5ln(3/4)+(5-3ln(81/64))/2