簡約-((1))/((2))ln^4(x^2)-((1))/((2))ln(x^2)

解

簡約

- \frac{( 1 )}{( 2 )} ln^{4} (x^{2}) - \frac{( 1 )}{( 2 )} ln (x^{2})

- ln^{4} (x^{2})

解答ステップ

- \frac{( 1 )}{( 2 )} ln^{4} (x^{2}) - \frac{( 1 )}{( 2 )} ln (x^{2})

対数の規則を適用する:

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

\frac{( 1 )}{( 2 )} ln^{4} (x^{2}) = ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}})

= - ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) - \frac{( 1 )}{( 2 )} ln (x^{2})

対数の規則を適用する:

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

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

= - ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) - ln ((x^{2})^{\frac{( 1 )}{( 2 )}})

- ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) - ln ((x^{2})^{\frac{( 1 )}{( 2 )}})

を書き換え

- (ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) + ln ((x^{2})^{\frac{( 1 )}{( 2 )}}))

= - (ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) + ln ((x^{2})^{\frac{( 1 )}{( 2 )}}))

対数の規則を適用する:

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

ln^{4} ((x^{2})^{\frac{( 1 )}{( 2 )}}) + ln ((x^{2})^{\frac{( 1 )}{( 2 )}}) = ln^{4} ((x^{2})^{\frac{1}{2}} (x^{2})^{\frac{1}{2}})

= - ln^{4} ((x^{2})^{\frac{1}{2}} (x^{2})^{\frac{1}{2}})

簡素化

(x^{2})^{\frac{( 1 )}{( 2 )}} (x^{2})^{\frac{( 1 )}{( 2 )}} : x^{2}

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