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-y))/(1-2y))

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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 - y )}{1 - 2 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} (ab)

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

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

Multiply fractions:

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

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

0.199 ln (\frac{0.0299}{0.075})

- lo g_{10} (3 x 1 0^{- 2})

- 4 ln (cos (x) x cos (x))

lo g_{7} (e^{- x} cos (π x))

ln^{1} (x) - ln^{6} (x)

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-y))/(1-2y))