What is ln(t^x(1-t)^{1-x}) ?

The solution to ln(t^x(1-t)^{1-x}) is xln(t)+(1-x)ln(1-t)

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simplify ln(t^x(1-t)^{1-x})

simplify

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

x ln (t) + (1 - x) ln (1 - t)

Solution steps

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

Apply log rule:

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

ln (t^{x} (1 - t)^{1 - x}) = ln (t^{x}) + ln ((1 - t)^{1 - x})

= ln (t^{x}) + ln ((- t + 1)^{- x + 1})

Apply log rule:

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

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

= x ln (t) + ln ((1 - t)^{1 - x})

Apply log rule:

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

ln ((1 - t)^{1 - x}) = (1 - x) ln (1 - t)

= x ln (t) + (1 - x) ln (1 - t)

ln (\frac{x ^{5} ( x - 7 ) ^{6}}{( x ^{2} + 8 ) ^{4}})

- \frac{8.31 \cdot 310}{9.65 \cdot 1 0 ^{4}} \cdot ln (\frac{140}{4})

ln (\frac{n + 1}{n})

ln ∣ x ∣ + ln ∣ 2 x + 1 ∣

ln \frac{( 399 e ^{- \frac{3}{80} x} )}{e ^{1584 x}}

What is ln(t^x(1-t)^{1-x}) ?
The solution to ln(t^x(1-t)^{1-x}) is xln(t)+(1-x)ln(1-t)