What is ln(4/3 θ^3e^{(-2θ)}) ?

The solution to ln(4/3 θ^3e^{(-2θ)}) is 2ln(2)-ln(3)+3ln(θ)-2θ

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simplify ln(4/3 θ^3e^{(-2θ)})

simplify

ln (\frac{4}{3} θ^{3} e^{(- 2 θ)})

2 ln (2) - ln (3) + 3 ln (θ) - 2 θ

Solution steps

ln (\frac{4}{3} θ^{3} e^{(- 2 θ)})

Apply log rule:

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

ln (\frac{4}{3} θ^{3} e^{(- 2 θ)}) = ln (\frac{4}{3}) + ln (θ^{3}) + ln (e^{(- 2 θ)})

= ln (\frac{4}{3}) + ln (θ^{3}) + ln (e^{(- 2 θ)})

Apply log rule:

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

ln (θ^{3}) = 3 ln (θ)

= ln (\frac{4}{3}) + 3 ln (θ) + ln (e^{- 2 θ})

Apply log rule:

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

ln (e^{- 2 θ}) = - 2 θ ln (e)

= ln (\frac{4}{3}) + 3 ln (θ) - 2 θ ln (e)

Apply log rule:

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

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

= ln (4) - ln (3) + 3 ln (θ) - 2 ln (e) θ

2 θ ln (e) = 2 θ

= ln (4) - ln (3) + 3 ln (θ) - 2 θ

ln (4) = 2 ln (2)

= 2 ln (2) - ln (3) + 3 ln (θ) - 2 θ

lo g_{2} (x) + lo g_{2} (7)

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

What is ln(4/3 θ^3e^{(-2θ)}) ?
The solution to ln(4/3 θ^3e^{(-2θ)}) is 2ln(2)-ln(3)+3ln(θ)-2θ