What is 1/(1/3-1)*ln((0.75-1)/(0.75*1/3-1)) ?

The solution to 1/(1/3-1)*ln((0.75-1)/(0.75*1/3-1)) is -(3(ln(-0.25)-ln(-(2.25)/3)))/(2)

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simplify 1/(1/3-1)ln((0.75-1)/(0.751/3-1))

simplify

\frac{1}{\frac{1}{3} - 1} \cdot ln (\frac{0.75 - 1}{0.75 \cdot \frac{1}{3} - 1})

- \frac{3 ( ln ( - 0.25 ) - ln ( - \frac{2.25}{3} ) )}{2}

Solution steps

\frac{1}{\frac{1}{3} - 1} ln (\frac{0.75 - 1}{0.75 \cdot \frac{1}{3} - 1})

Apply log rule:

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

ln (\frac{0.75 - 1}{0.75 \cdot \frac{1}{3} - 1}) = ln (0.75 - 1) - ln (0.75 \cdot \frac{1}{3} - 1)

= \frac{1}{\frac{1}{3} - 1} (ln (0.75 - 1) - ln (0.75 \cdot \frac{1}{3} - 1))

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( ln ( 0.75 - 1 ) - ln ( 0.75 \cdot \frac{1}{3} - 1 ) )}{\frac{1}{3} - 1}

1 \cdot (ln (0.75 - 1) - ln (0.75 \cdot \frac{1}{3} - 1)) = ln (- 0.25) - ln (0.75 \cdot \frac{1}{3} - 1)

= \frac{ln ( - 0.25 ) - ln ( 0.75 \cdot \frac{1}{3} - 1 )}{\frac{1}{3} - 1}

Join

\frac{1}{3} - 1 : - \frac{2}{3}

= \frac{ln ( - 0.25 ) - ln ( 0.75 \cdot \frac{1}{3} - 1 )}{- \frac{2}{3}}

ln (0.75 \cdot \frac{1}{3} - 1) = ln (- \frac{2.25}{3})

= \frac{ln ( - 0.25 ) - ln ( - \frac{2.25}{3} )}{- \frac{2}{3}}

Apply the fraction rule:

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

= - \frac{ln ( - 0.25 ) - ln ( - \frac{2.25}{3} )}{\frac{2}{3}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

\frac{ln ( - 0.25 ) - ln ( - \frac{2.25}{3} )}{\frac{2}{3}} = \frac{( ln ( - 0.25 ) - ln ( - \frac{2.25}{3} ) ) \cdot 3}{2}

= - \frac{( ln ( - 0.25 ) - ln ( - \frac{2.25}{3} ) ) \cdot 3}{2}

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What is 1/(1/3-1)*ln((0.75-1)/(0.75*1/3-1)) ?
The solution to 1/(1/3-1)*ln((0.75-1)/(0.75*1/3-1)) is -(3(ln(-0.25)-ln(-(2.25)/3)))/(2)