What is ln(x^y*(1-x)^{n-y}) ?

The solution to ln(x^y*(1-x)^{n-y}) is yln(x)+(n-y)ln(1-x)

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simplify ln(x^y*(1-x)^{n-y})

simplify

ln (x^{y} \cdot (1 - x)^{n - y})

y ln (x) + (n - y) ln (1 - x)

Solution steps

ln (x^{y} (1 - x)^{n - y})

Apply log rule:

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

ln (x^{y} (1 - x)^{n - y}) = ln (x^{y}) + ln ((1 - x)^{n - y})

= ln (x^{y}) + ln ((- x + 1)^{- y + n})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (x^{y}) = y ln (x)

= y ln (x) + ln ((1 - x)^{n - y})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((1 - x)^{n - y}) = (n - y) ln (1 - x)

= y ln (x) + (n - y) ln (1 - x)

\frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣

lo g_{e} (\frac{2}{3} w)

2 ln ∣ x + 2 ∣ - ln ∣ x - 4 ∣ + C

ln (θ^{y} (1 - θ^{n - y}))

What is ln(x^y*(1-x)^{n-y}) ?
The solution to ln(x^y*(1-x)^{n-y}) is yln(x)+(n-y)ln(1-x)