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

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

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

simplify

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

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

Decimal Notation

1.21231 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

2 ln (3) = ln (3^{2})

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

Apply log rule:

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

ln (\frac{2}{3}) + ln (3^{2}) = ln (3^{2} \cdot \frac{2}{3})

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

\frac{2}{3} \cdot 3^{2} = 6

= \frac{3}{2} + ln (6) - 3 ln (2)

Apply log rule:

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

3 ln (2) = ln (2^{3})

= \frac{3}{2} + ln (6) - ln (2^{3})

Apply log rule:

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

ln (6) - ln (2^{3}) = ln (\frac{6}{2 ^{3}})

= \frac{3}{2} + ln (\frac{6}{2 ^{3}})

\frac{6}{2 ^{3}} = \frac{3}{4}

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

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