What is ln((e^{-θ}θ^x)/(x!)) ?

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

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simplify ln((e^{-θ}θ^x)/(x!))

simplify

ln (\frac{e ^{- θ} θ ^{x}}{x !})

- θ + x ln (θ) - ln (x!)

Solution steps

ln (\frac{e ^{- θ} θ ^{x}}{x !})

Apply log rule:

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

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

= ln (e^{- θ}) + ln (θ^{x}) - ln (x!)

Apply log rule:

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

ln (e^{- θ}) = - θ ln (e)

= - θ ln (e) + ln (θ^{x}) - ln (x!)

Apply log rule:

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

ln (θ^{x}) = x ln (θ)

= - θ ln (e) + x ln (θ) - ln (x!)

θ ln (e) = θ

= - θ + x ln (θ) - ln (x!)

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