What is ln((3x^2)/θ*e^{-(3x^2)/θ}) ?

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

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simplify ln((3x^2)/θ*e^{-(3x^2)/θ})

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

ln (x^{2} - 2 x - 24) - ln (x + 4)

- lo g_{10} (4 \cdot 1 0^{- 6})

- ln (\frac{4}{3})

\frac{1}{2} ln (\frac{3}{4}) - \frac{1}{2} ln (\frac{1}{2})

\frac{1}{3} lo g_{2} (3)

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