簡約 (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5}

解

簡約

(4 ln (x - 6) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

ln^{\frac{1}{5}} (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})

解答ステップ

(4 ln (x - 6) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

4 ln (x - 6) = ln ((x - 6)^{4})

= (ln ((x - 6)^{4}) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

対数の規則を適用する:

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

ln ((x - 6)^{4}) + ln (x^{2} + 6) = ln ((x - 6)^{4} (x^{2} + 6))

= (ln ((x - 6)^{4} (x^{2} + 6)) - 7 ln (x + 3))^{\frac{1}{5}}

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

7 ln (x + 3) = ln ((x + 3)^{7})

= (ln ((x - 6)^{4} (x^{2} + 6)) - ln ((x + 3)^{7}))^{\frac{1}{5}}

対数の規則を適用する:

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

ln ((x - 6)^{4} (x^{2} + 6)) - ln ((x + 3)^{7}) = ln (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})

= ln^{\frac{1}{5}} (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})