What is ln((3x^2)/θ e^{(-x^3)/θ}) ?

The solution to ln((3x^2)/θ e^{(-x^3)/θ}) is ln(3)+2ln(x)-ln(θ)-(x^3)/θ

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simplify ln((3x^2)/θ e^{(-x^3)/θ})

simplify

ln (\frac{3 x ^{2}}{θ} e^{\frac{- x ^{3}}{θ}})

ln (3) + 2 ln (x) - ln (θ) - \frac{x ^{3}}{θ}

Solution steps

ln (\frac{3 x ^{2}}{θ} e^{\frac{- x ^{3}}{θ}})

Apply log rule:

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

ln (\frac{3 x ^{2}}{θ} e^{\frac{- x ^{3}}{θ}}) = ln (\frac{3 x ^{2}}{θ}) + ln (e^{\frac{- x ^{3}}{θ}})

= ln (\frac{3 x ^{2}}{θ}) + ln (e^{\frac{- x ^{3}}{θ}})

Apply log rule:

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

ln (e^{\frac{- x ^{3}}{θ}}) = \frac{- x ^{3}}{θ} ln (e)

= ln (\frac{3 x ^{2}}{θ}) + \frac{- x ^{3}}{θ} ln (e)

Apply log rule:

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

ln (\frac{3 x ^{2}}{θ}) = ln (3 x^{2}) - ln (θ)

= ln (3 x^{2}) - ln (θ) + ln (e) \frac{- x ^{3}}{θ}

Apply log rule:

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

ln (3 x^{2}) = ln (3) + ln (x^{2})

= ln (3) + ln (x^{2}) - ln (θ) + ln (e) \frac{- x ^{3}}{θ}

Apply log rule:

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

ln (x^{2}) = 2 ln (x)

= ln (3) + 2 ln (x) - ln (θ) + \frac{- x ^{3}}{θ} ln (e)

\frac{- x ^{3}}{θ} ln (e) = - \frac{x ^{3}}{θ}

= ln (3) + 2 ln (x) - ln (θ) - \frac{x ^{3}}{θ}

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\frac{2000 ln ( \frac{11}{125} )}{ln ( 5 )}

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What is ln((3x^2)/θ e^{(-x^3)/θ}) ?
The solution to ln((3x^2)/θ e^{(-x^3)/θ}) is ln(3)+2ln(x)-ln(θ)-(x^3)/θ