What is ln(θe^θx^{-θ-1}) ?

The solution to ln(θe^θx^{-θ-1}) is ln(θ)+θ+ln(x)(-θ-1)

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simplify ln(θe^θx^{-θ-1})

simplify

ln (θ e^{θ} x^{- θ - 1})

ln (θ) + θ + ln (x) (- θ - 1)

Solution steps

ln (θ e^{θ} x^{- θ - 1})

Apply log rule:

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

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

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

Apply log rule:

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

ln (e^{θ}) = θ ln (e)

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

Apply log rule:

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

ln (x^{- θ - 1}) = (- θ - 1) ln (x)

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

θ ln (e) = θ

= ln (θ) + θ + ln (x) (- θ - 1)

lo g_{10} (x^{3} \cdot y^{2})

lo g_{2} ((3 x)^{3})

lo g_{6} (7 x^{2})

lo g_{4} (\frac{x ^{2} - 2 x - 3}{x ^{2} - 3 x})

ln (x^{8} y^{3} z)

What is ln(θe^θx^{-θ-1}) ?
The solution to ln(θe^θx^{-θ-1}) is ln(θ)+θ+ln(x)(-θ-1)