What is ln(2y)-ln(y)-ln(1-2y)-ln(1-y) ?

The solution to ln(2y)-ln(y)-ln(1-2y)-ln(1-y) is ln(2/((1-2y)(1-y)))

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simplify ln(2y)-ln(y)-ln(1-2y)-ln(1-y)

simplify

ln (2 y) - ln (y) - ln (1 - 2 y) - ln (1 - y)

ln (\frac{2}{( 1 - 2 y ) ( 1 - y )})

Solution steps

ln (2 y) - ln (y) - ln (1 - 2 y) - ln (1 - y)

Apply log rule:

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

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

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

Cancel the common factor:

y

= ln (2) - ln (1 - 2 y) - ln (1 - y)

Apply log rule:

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

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

= ln (\frac{2}{1 - 2 y}) - ln (1 - y)

Apply log rule:

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

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

= ln (\frac{\frac{2}{1 - 2 y}}{1 - y})

Apply the fraction rule:

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

= ln (\frac{2}{( 1 - 2 y ) ( 1 - y )})

100 ln (\frac{55}{52})

ln (\frac{6 x e ^{5}}{7 y e ^{2}})

0.0296 \cdot 46 e^{(\frac{l n ( 263 ) - l n ( 46 )}{30})}

- lo g_{10} (6.652 \cdot 1 0^{- 5})

What is ln(2y)-ln(y)-ln(1-2y)-ln(1-y) ?
The solution to ln(2y)-ln(y)-ln(1-2y)-ln(1-y) is ln(2/((1-2y)(1-y)))