What is 3ln(2)-3ln(2+1) ?

The solution to 3ln(2)-3ln(2+1) is ln(8/27)

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simplify 3ln(2)-3ln(2+1)

simplify

3 ln (2) - 3 ln (2 + 1)

ln (\frac{8}{27})

Decimal Notation

- 1.21639 \dots

Solution steps

3 ln (2) - 3 ln (2 + 1)

Apply log rule:

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

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

= ln (2^{3}) - 3 ln (2 + 1)

Apply log rule:

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

3 ln (2 + 1) = ln ((2 + 1)^{3})

= ln (2^{3}) - ln ((2 + 1)^{3})

Apply log rule:

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

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

= ln (\frac{2 ^{3}}{( 2 + 1 ) ^{3}})

\frac{2 ^{3}}{( 2 + 1 ) ^{3}} = \frac{8}{27}

= ln (\frac{8}{27})

ln (\frac{a ^{2}}{b ^{- 3} c ^{2}})

ln (\frac{10000}{8600})

ln (x^{3}) + ln (y)

ln (\frac{1 + 8}{2 + 16})

- \frac{2 ln ( 10 ) + ln ( 97 )}{6}