What is (ln(9)-6ln(6)+6ln(2))/2 ?

The solution to (ln(9)-6ln(6)+6ln(2))/2 is -2ln(3)

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simplify (ln(9)-6ln(6)+6ln(2))/2

simplify

\frac{ln ( 9 ) - 6 ln ( 6 ) + 6 ln ( 2 )}{2}

- 2 ln (3)

Decimal Notation

- 2.19722 \dots

Solution steps

\frac{ln ( 9 ) - 6 ln ( 6 ) + 6 ln ( 2 )}{2}

Apply log rule:

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

6 ln (6) = ln (6^{6})

= \frac{ln ( 9 ) - ln ( 6 ^{6} ) + 6 ln ( 2 )}{2}

Apply log rule:

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

ln (9) - ln (6^{6}) = ln (\frac{9}{6 ^{6}})

= \frac{ln ( \frac{9}{6 ^{6}} ) + 6 ln ( 2 )}{2}

\frac{9}{6 ^{6}} = \frac{1}{5184}

= \frac{ln ( \frac{1}{5184} ) + 6 ln ( 2 )}{2}

Apply log rule:

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

6 ln (2) = ln (2^{6})

= \frac{ln ( \frac{1}{5184} ) + ln ( 2 ^{6} )}{2}

Apply log rule:

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

ln (\frac{1}{5184}) + ln (2^{6}) = ln (2^{6} \cdot \frac{1}{5184})

= \frac{ln ( 2 ^{6} \cdot \frac{1}{5184} )}{2}

\frac{1}{5184} \cdot 2^{6} = \frac{1}{81}

= \frac{ln ( \frac{1}{81} )}{2}

Simplify

ln (\frac{1}{81}) : - 4 ln (3)

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

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

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

Divide the numbers:

\frac{4}{2} = 2

= - 2 ln (3)

ln (\frac{2 i - 3}{2 i + 3})

ln (5.6 \cdot 1 0^{11})

lo g_{2} (e^{9}) - lo g_{2} (e^{6})

lo g_{5} (8 \cdot 11 \cdot 11 \cdot 3^{3})

5 (- ln ∣ 1 - 4 ∣ + ln ∣ 1 - 5 ∣) + y + C

What is (ln(9)-6ln(6)+6ln(2))/2 ?
The solution to (ln(9)-6ln(6)+6ln(2))/2 is -2ln(3)