簡約 ln((pN^{1/r})/(K^{1/r)(-p+1)})

解

簡約

ln (\frac{p N ^{\frac{1}{r}}}{K ^{\frac{1}{r}} ( - p + 1 )})

ln (p) + \frac{ln ( N ) - ln ( K )}{r} - ln (- p + 1)

解答ステップ

ln (\frac{p N ^{\frac{1}{r}}}{K ^{\frac{1}{r}} ( - p + 1 )})

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{p N ^{\frac{1}{r}}}{K ^{\frac{1}{r}} ( - p + 1 )}) = ln (p N^{\frac{1}{r}}) - ln (K^{\frac{1}{r}} (- p + 1))

= ln (N^{\frac{1}{r}} p) - ln (K^{\frac{1}{r}} (- p + 1))

対数の規則を適用する:

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

ln (p N^{\frac{1}{r}}) = ln (p) + ln (N^{\frac{1}{r}}), ln (K^{\frac{1}{r}} (- p + 1)) = ln (K^{\frac{1}{r}}) + ln (- p + 1)

= ln (p) + ln (N^{\frac{1}{r}}) - (ln (- p + 1) + ln (K^{\frac{1}{r}}))

対数の規則を適用する:

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

ln (N^{\frac{1}{r}}) = \frac{1}{r} ln (N)

= ln (p) + \frac{1}{r} ln (N) - (ln (K^{\frac{1}{r}}) + ln (- p + 1))

対数の規則を適用する:

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

ln (K^{\frac{1}{r}}) = \frac{1}{r} ln (K)

= ln (p) + \frac{1}{r} ln (N) - (\frac{1}{r} ln (K) + ln (- p + 1))

\frac{1}{r} ln (N) = \frac{ln ( N )}{r}

- (\frac{1}{r} ln (K) + ln (- p + 1)) = - (\frac{ln ( K )}{r} + ln (- p + 1))

= ln (p) + \frac{ln ( N )}{r} - (ln (- p + 1) + \frac{ln ( K )}{r})

- (\frac{ln ( K )}{r} + ln (- p + 1)) : - \frac{ln ( K )}{r} - ln (- p + 1)

= ln (p) + \frac{ln ( N )}{r} - \frac{ln ( K )}{r} - ln (- p + 1)

分数を組み合わせる

\frac{ln ( N )}{r} - \frac{ln ( K )}{r} : \frac{ln ( N ) - ln ( K )}{r}

= ln (p) + \frac{- ln ( K ) + ln ( N )}{r} - ln (- p + 1)

= ln (p) + \frac{ln ( N ) - ln ( K )}{r} - ln (- p + 1)