What is ln(1/2 e^{-x^2}) ?

The solution to ln(1/2 e^{-x^2}) is -x^2-ln(2)

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simplify ln(1/2 e^{-x^2})

simplify

ln (\frac{1}{2} e^{- x^{2}})

- x^{2} - ln (2)

Solution steps

ln (\frac{1}{2} e^{- x^{2}})

Apply log rule:

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

ln (\frac{1}{2} e^{- x^{2}}) = ln (\frac{1}{2}) + ln (e^{- x^{2}})

= ln (\frac{1}{2}) + ln (e^{- x^{2}})

Apply log rule:

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

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

= ln (\frac{1}{2}) - x^{2} ln (e)

Apply log rule:

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

ln (\frac{1}{2}) = ln (1) - ln (2)

= ln (1) - ln (2) - ln (e) x^{2}

ln (1) = 0

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

= 0 - ln (2) - x^{2}

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

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

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