What is ln(nθ^{2n}xe^{-θnx}) ?

The solution to ln(nθ^{2n}xe^{-θnx}) is ln(n)+2nln(θ)+ln(x)-nθx

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simplify ln(nθ^{2n}xe^{-θnx})

simplify

ln (n θ^{2 n} x e^{- θ n x})

ln (n) + 2 n ln (θ) + ln (x) - n θ x

Solution steps

ln (n θ^{2 n} x e^{- θ n x})

Apply log rule:

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

ln (n θ^{2 n} x e^{- θ n x}) = ln (n) + ln (θ^{2 n}) + ln (x) + ln (e^{- θ n x})

= ln (n) + ln (θ^{2 n}) + ln (x) + ln (e^{- n θ x})

Apply log rule:

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

ln (θ^{2 n}) = 2 n ln (θ)

= ln (n) + 2 n ln (θ) + ln (x) + ln (e^{- θ n x})

Apply log rule:

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

ln (e^{- θ n x}) = - θ n x ln (e)

= ln (n) + 2 n ln (θ) + ln (x) - θ n x ln (e)

θ n x ln (e) = n θ x

= ln (n) + 2 n ln (θ) + ln (x) - n θ x

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What is ln(nθ^{2n}xe^{-θnx}) ?
The solution to ln(nθ^{2n}xe^{-θnx}) is ln(n)+2nln(θ)+ln(x)-nθx