What is ln(1/((r-1)!)θ^{-r}x^{(r-1)}e^{(-xθ-1)}) ?

The solution to ln(1/((r-1)!)θ^{-r}x^{(r-1)}e^{(-xθ-1)}) is -rln(θ)+ln(x)(r-1)-ln((r-1)!)-θx-1

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simplify ln(1/((r-1)!)θ^{-r}x^{(r-1)}e^{(-xθ-1)})

simplify

ln (\frac{1}{( r - 1 ) !} θ^{- r} x^{(r - 1)} e^{(- x θ - 1)})

- r ln (θ) + ln (x) (r - 1) - ln ((r - 1)!) - θ x - 1

Solution steps

ln (\frac{1}{( r - 1 ) !} θ^{- r} x^{(r - 1)} e^{(- x θ - 1)})

Apply log rule:

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

ln (\frac{1}{( r - 1 ) !} θ^{- r} x^{(r - 1)} e^{(- x θ - 1)}) = ln (\frac{1}{( r - 1 ) !}) + ln (θ^{- r}) + ln (x^{(r - 1)}) + ln (e^{(- x θ - 1)})

= ln (\frac{1}{( r - 1 ) !}) + ln (θ^{- r}) + ln (x^{(r - 1)}) + ln (e^{(- θ x - 1)})

Apply log rule:

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

ln (θ^{- r}) = - r ln (θ)

= ln (\frac{1}{( r - 1 ) !}) - r ln (θ) + ln (x^{r - 1}) + ln (e^{- x θ - 1})

Apply log rule:

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

ln (x^{r - 1}) = (r - 1) ln (x)

= ln (\frac{1}{( r - 1 ) !}) - r ln (θ) + (r - 1) ln (x) + ln (e^{- x θ - 1})

Apply log rule:

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

ln (e^{- x θ - 1}) = (- x θ - 1) ln (e)

= ln (\frac{1}{( r - 1 ) !}) - r ln (θ) + (r - 1) ln (x) + (- x θ - 1) ln (e)

Apply log rule:

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

ln (\frac{1}{( r - 1 ) !}) = ln (1) - ln ((r - 1)!)

= ln (1) - ln ((r - 1)!) - r ln (θ) + ln (x) (r - 1) + ln (e) (- θ x - 1)

ln (1) = 0

(- x θ - 1) ln (e) = - θ x - 1

= 0 - ln ((r - 1)!) - r ln (θ) + ln (x) (r - 1) + (- θ x - 1)

Remove parentheses:

(- a) = - a

= 0 - ln ((r - 1)!) - r ln (θ) + (r - 1) ln (x) - x θ - 1

Group like terms

= - r ln (θ) + ln (x) (r - 1) - ln ((r - 1)!) - θ x + 0 - 1

Add/Subtract the numbers:

0 - 1 = - 1

= - r ln (θ) + ln (x) (r - 1) - ln ((r - 1)!) - θ x - 1

ln (4) + 4 ln (3)

2400 + \frac{1}{7.13 \cdot 1 0 ^{- 4}} \cdot ln (\frac{1}{0.9})

- ln (x) + 2 ln (x + 1)

ln (\frac{b}{x}) + ln (\frac{a}{x})

ln (x - 3) + ln (x + 5) - 2 ln (x)

What is ln(1/((r-1)!)θ^{-r}x^{(r-1)}e^{(-xθ-1)}) ?
The solution to ln(1/((r-1)!)θ^{-r}x^{(r-1)}e^{(-xθ-1)}) is -rln(θ)+ln(x)(r-1)-ln((r-1)!)-θx-1