簡約 ln(x/(u^2)e^{-(x^2)/(2u^2)})

解

簡約

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

ln (x) - 2 ln (u) - \frac{x ^{2}}{2 u ^{2}}

解答ステップ

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

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

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