What is ln(t^y*(e^{-t})/(y!)) ?

The solution to ln(t^y*(e^{-t})/(y!)) is yln(t)-t-ln(y!)

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simplify ln(t^y*(e^{-t})/(y!))

simplify

ln (t^{y} \cdot \frac{e ^{- t}}{y !})

y ln (t) - t - ln (y!)

Solution steps

ln (t^{y} \frac{e ^{- t}}{y !})

Apply log rule:

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

ln (t^{y} \frac{e ^{- t}}{y !}) = ln (t^{y}) + ln (\frac{e ^{- t}}{y !})

= ln (t^{y}) + ln (\frac{e ^{- t}}{y !})

Apply log rule:

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

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

= y ln (t) + ln (\frac{e ^{- t}}{y !})

Apply log rule:

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

ln (\frac{e ^{- t}}{y !}) = ln (e^{- t}) - ln (y!)

= y ln (t) + + ln (e^{- t}) - ln (y!)

Apply log rule:

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

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

= y ln (t) - t ln (e) - ln (y!)

t ln (e) = t

= y ln (t) - t - ln (y!)

- \frac{1000 ln ( \frac{11600}{19971} )}{3}

ln (\frac{9}{7}) \cdot \frac{500}{6}

3 lo g_{b} (x) - 6 lo g_{b} (z) - 4 lo g_{b} (w)

4 + 2 lo g_{5} (x) - \frac{1}{3} lo g_{5} (y)

- ln (\frac{2 l + d}{l}) + ln (\frac{l + d}{l}) + ln (\frac{l + d}{d})

What is ln(t^y*(e^{-t})/(y!)) ?
The solution to ln(t^y*(e^{-t})/(y!)) is yln(t)-t-ln(y!)