What is ln(2/3)+1/6 ln(2/9)+1/3 ln(4/9) ?

The solution to ln(2/3)+1/6 ln(2/9)+1/3 ln(4/9) is ln((2\sqrt[3]{4}\sqrt[6]{2})/9)

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simplify ln(2/3)+1/6 ln(2/9)+1/3 ln(4/9)

simplify

ln (\frac{2}{3}) + \frac{1}{6} ln (\frac{2}{9}) + \frac{1}{3} ln (\frac{4}{9})

Decimal Notation

- 0.92645 \dots

Solution steps

ln (\frac{2}{3}) + \frac{1}{6} ln (\frac{2}{9}) + \frac{1}{3} ln (\frac{4}{9})

Apply log rule:

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

\frac{1}{6} ln (\frac{2}{9}) = ln ((\frac{2}{9})^{\frac{1}{6}})

= ln (\frac{2}{3}) + ln ((\frac{2}{9})^{\frac{1}{6}}) + \frac{1}{3} ln (\frac{4}{9})

Apply log rule:

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

ln (\frac{2}{3}) + ln ((\frac{2}{9})^{\frac{1}{6}}) = ln (\frac{2}{3} (\frac{2}{9})^{\frac{1}{6}})

= ln (\frac{2}{3} (\frac{2}{9})^{\frac{1}{6}}) + \frac{1}{3} ln (\frac{4}{9})

Apply log rule:

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

\frac{1}{3} ln (\frac{4}{9}) = ln ((\frac{4}{9})^{\frac{1}{3}})

Apply log rule:

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

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What is ln(2/3)+1/6 ln(2/9)+1/3 ln(4/9) ?
The solution to ln(2/3)+1/6 ln(2/9)+1/3 ln(4/9) is ln((2\sqrt[3]{4}\sqrt[6]{2})/9)