What is 3*ln((2/3)^3)-ln(1-2/3) ?

The solution to 3*ln((2/3)^3)-ln(1-2/3) is ln(512/6561)

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simplify 3*ln((2/3)^3)-ln(1-2/3)

simplify

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

ln (\frac{512}{6561})

Decimal Notation

- 2.55057 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{( ( \frac{2}{3} ) ^{3} ) ^{3}}{1 - \frac{2}{3}} = \frac{512}{6561}

= ln (\frac{512}{6561})

ln (1.243) - ln (1.242) - 0.275

\frac{1}{2} ln (x) - ln (4)

ln (3 \cdot 4)

\frac{1}{2} ln (x) - ln (3)

6 lo g_{2} (u) - 24 lo g_{2} (v)

What is 3*ln((2/3)^3)-ln(1-2/3) ?
The solution to 3*ln((2/3)^3)-ln(1-2/3) is ln(512/6561)