What is ln(((e^{-u}u^x))/(x!)) ?

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

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simplify ln(((e^{-u}u^x))/(x!))

simplify

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

- u + x ln (u) - ln (x!)

Solution steps

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

Apply log rule:

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

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

= ln (e^{- u} u^{x}) - ln (x!)

Apply log rule:

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

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

= ln (e^{- u}) + ln (u^{x}) - ln (x!)

Apply log rule:

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

ln (e^{- u}) = - u ln (e)

= - u ln (e) + ln (u^{x}) - ln (x!)

Apply log rule:

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

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

= - u ln (e) + x ln (u) - ln (x!)

u ln (e) = u

= - u + x ln (u) - ln (x!)

lo g_{4} (\frac{1}{126})

ln (\frac{85}{28.05})

ln (\frac{x}{1.5})

24 lo g_{5} (c) + lo g_{5} (ab)

ln (\frac{70}{80})

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