簡約 ln((x^2)/(esqrt(x)cos(x)))

解

簡約

ln (\frac{x ^{2}}{e x cos ( x )})

2 ln (x) - 1 - ln (x) - ln (cos (x))

解答ステップ

ln (\frac{x ^{2}}{e x cos ( x )})

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

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

対数の規則を適用する:

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

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

= 2 ln (x) - ln (e x cos (x))

対数の規則を適用する:

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

ln (e x cos (x)) = ln (e) + ln (x) + ln (cos (x))

= 2 ln (x) - (ln (x) + ln (cos (x)) + ln (e))

対数の規則を適用する:

lo g_{a} (a) = 1

= 2 ln (x) - (ln (x) + ln (cos (x)) + 1)

- (1 + ln (x) + ln (cos (x))) : - 1 - ln (x) - ln (cos (x))

= 2 ln (x) - 1 - ln (x) - ln (cos (x))