3.4-bo'lim: Logarifmlarni soddalashtirish
Quyidagi logarifmlarning har birini soddalashtiring.
- \(\ln {x^3}{y^4}{z^5}\)
Yechim
Bu erda soddalashtirish 1-dan 7-gachasi xususiyatdan foydalanishni anglatadi Logarifm xossalari imkon qadar tez-tez bo'lim. Bu xususiyatlardan boshqa foydalana olmaganingizda bajarasiz.
5-xususiyat ikki funktsiyadan ortiq mahsulotlarga kengaytirilishi mumkin, shuning uchun,
\[\begin{align*}\ln {x^3}{y^4}{z^5} & = \ln {x^3} + \ln {y^4} + \ln {z^5}\\ & = 3\ln x + 4\ln y + 5\ln z\end{align*}\] - \({\log _3}\left( {\frac{{9{x^4}}}{{\sqrt y }}} \right)\)
Yechim
6-xususiyatdan foydalanishda siz olib tashlagan logarifm uning argumenti sifatida maxrajni o'z ichiga olgan logarifm ekanligiga ishonch hosil qiling. Shuni ham yodda tutingki, men birinchi bosqichda ildizni ko'rsatkichlarga aylantiraman, chunki bu keyingi bosqichda bajarilishi kerak.
\[\begin{align*}{\log _3}\left( {\frac{{9{x^4}}}{{\sqrt y }}} \right) & = {\log _3}9{x^4} - {\log _3}{y^{\frac{1}{2}}}\\ & = {\log _3}9 + {\log _3}{x^4} - {\log _3}{y^{\frac{1}{2}}}\\ & = 2 + 4{\log _3}x - \frac{1}{2}{\log _3}y\end{align*}\]Iloji bo'lsa, birinchi davrdagi kabi jurnallarni baholang.
- \(\log \left( {\frac{{{x^2} + {y^2}}}{{{{\left( {x - y} \right)}^3}}}} \right)\)
Yechim
Ushbu muammoning mohiyati asosan 7-mulkdan to'g'ri foydalanishdir.
\[\begin{align*}\log \left( {\frac{{{x^2} + {y^2}}}{{{{\left( {x - y} \right)}^3}}}} \right) & = \log \left( {{x^2} + {y^2}} \right) - \log {\left( {x - y} \right)^3}\\ & = \log \left( {{x^2} + {y^2}} \right) - 3\log \left( {x - y} \right)\end{align*}\]7-xususiyatdan ikkinchi atamada foydalanishingiz mumkin, chunki BUTUN atama 3 ga ko'tarildi, lekin birinchi logarifmda faqat individual shartlar kvadratga aylangan va umuman atama emas, shuning uchun 2 lar turgan joyida qolishi kerak!