What is ln((θ^xe^{-θ})/(x!)) ?

The solution to ln((θ^xe^{-θ})/(x!)) is xln(θ)-θ-ln(x!)

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simplify ln((θ^xe^{-θ})/(x!))

simplify

ln (\frac{θ ^{x} e ^{- θ}}{x !})

x ln (θ) - θ - ln (x!)

Solution steps

ln (\frac{θ ^{x} e ^{- θ}}{x !})

Apply log rule:

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

ln (\frac{θ ^{x} e ^{- θ}}{x !}) = ln (θ^{x} e^{- θ}) - ln (x!)

= ln (e^{- θ} θ^{x}) - ln (x!)

Apply log rule:

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

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

= ln (θ^{x}) + ln (e^{- θ}) - ln (x!)

Apply log rule:

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

ln (θ^{x}) = x ln (θ)

= x ln (θ) + ln (e^{- θ}) - ln (x!)

Apply log rule:

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

ln (e^{- θ}) = - θ ln (e)

= x ln (θ) - θ ln (e) - ln (x!)

θ ln (e) = θ

= x ln (θ) - θ - ln (x!)

8 ln (2 e)

lo g_{10} (o^{x i} (1 - o)^{1 - x i})

ln (\frac{155}{100})

ln (\frac{12 x}{4 x})

ln (\frac{1}{\frac{x}{y}})

What is ln((θ^xe^{-θ})/(x!)) ?
The solution to ln((θ^xe^{-θ})/(x!)) is xln(θ)-θ-ln(x!)