3.3-bo'lim: Logarifm xususiyatlari
Quyidagi formulalarni to‘ldiring.
- \({\log _b}b = \)
Yechim
\[{\log _b}b = 1\hspace{0.5in}{\rm{because}}\hspace{0.5in}{b^1} = b\] - \({\log _b}1 = \)
Yechim
\[{\log _b}1 = 0\hspace{0.5in}{\rm{because}}\hspace{0.5in}{b^0} = 1\] - \({\log _b}{b^x} = \)
Yechim
\[{\log _b}{b^x} = x\] - \({b^{{{\log }_b}x}} = \)
Yechim
\[{b^{{{\log }_b}x}} = x\] - \({\log _b}xy = \)
Yechim
\[{\log _b}xy = {\log _b}x + {\log _b}y\]SO'MLAR YO FAQLARI UCHUN BUNDAY MULK YO'Q!!!!!
\[\begin{align*}{\log _b}\left( {x + y} \right) & \ne {\log _b}x + {\log _b}y\\ {\log _b}\left( {x - y} \right) & \ne {\log _b}x - {\log _b}y\end{align*}\] - \({\log _b}\left( {\frac{x}{y}} \right) = \)
Yechim
\[{\log _b}\left( {\frac{x}{y}} \right) = {\log _b}x - {\log _b}y\]SO'MLAR YO FAQLARI UCHUN BUNDAY MULK YO'Q!!!!!
\[\begin{align*}{\log _b}\left( {x + y} \right) &\ne {\log _b}x + {\log _b}y\\ {\log _b}\left( {x - y} \right) &\ne {\log _b}x - {\log _b}y\end{align*}\] - \({\log _b}\left( {{x^r}} \right) = \)
Yechim
\[{\log _b}\left( {{x^r}} \right) = r{\log _b}x\]E'tibor bering, bu holda ko'rsatkich logarifmning BUTUN argumentida bo'lishi kerak. Masalan,
\[{\log _b}{\left( {x + y} \right)^2} = 2{\log _b}\left( {x + y} \right)\]Biroq,
\[{\log _b}\left( {{x^2} + {y^2}} \right) \ne 2{\log _b}\left( {x + y} \right)\] - Logarifmlar uchun asosiy formulaning o‘zgarishini yozing.
Yechim
Logarifmlar uchun asosiy formulaning o'zgarishi,
\[{\log _b}x = \frac{{{{\log }_a}x}}{{{{\log }_a}b}}\]Bu asosiy formulaning eng umumiy o'zgarishi bo'lib, \(b\) bazasidan \(a\) bazasiga aylanadi. Biroq, asosiy formulani o'zgartirishdan foydalanishning odatiy sababi shundaki, siz osongina hisoblab bo'lmaydigan bazada joylashgan logarifmning qiymatini hisoblashingiz mumkin. Asosiy formulani o'zgartirish, siz logarifmni hisoblashingiz mumkin bo'lgan logarifm shaklida yozishingiz mumkinligini anglatadi. Asosiy formulalarning ikkita eng keng tarqalgan o'zgarishi
\[{\log _b}x = \frac{{\ln x}}{{\ln b}}\hspace{0.5in}{\rm{and}}\hspace{0.5in}{\rm{lo}}{{\rm{g}}_{\rm{b}}}x = \frac{{\log x}}{{\log b}}\]Haqiqatan ham, siz ko'pincha asosiy formulaning o'zgarishi sifatida ro'yxatga olingan birini yoki boshqasini ko'rasiz!
dagi muammolarda Logarifmning asosiy funktsiyalari bo'limda siz bir nechta logarifmlarning qiymatini hisobladingiz, lekin siz buni asosiy formulani o'zgartirmasdan qilishingiz mumkin, chunki barcha argumentlar baza nuqtai nazaridan bir darajaga yozilishi mumkin. Masalan,
\[{\log _7}49 = 2\hspace{0.5in}{\rm{because}}\hspace{0.5in}{7^2} = 49\]Biroq, bu faqat ishlaydi, chunki 49 ni 7 ning kuchi sifatida yozish mumkin! \({\log _7}50\) ni hisoblash uchun bizga asosiy formulani o'zgartirish kerak bo'ladi.
\[{\log _7}50 = \frac{{\ln 50}}{{\ln 7}} = \frac{{3.91202300543}}{{1.94591014906}} = 2.0103821378\]OR
\[{\log _7}50 = \frac{{\log 50}}{{\log 7}} = \frac{{1.69897000434}}{{0.845098040014}} = 2.0103821378\]Shunday qilib, qaysi birini ishlatishimiz muhim emas, siz bir xil javob olasiz.
Shuni ham yodda tutingki, agar xohlasak, biz \({\log _7}49\) da asosiy formulani o'zgartirishimiz mumkin.
\[{\log _7}49 = \frac{{\ln 49}}{{\ln 7}} = \frac{{3.89182029811}}{{1.94591014906}} = 2\]Biroq, bu juda ko'p ish va bu bilan kurashishning eng yaxshi usuli emas.
- Logarifmning sohasi nima?
Yechim
\(y = {\log _b}x\) domeni \(x > 0\). Boshqacha qilib aytganda, logarifmaga nol yoki manfiy raqamni kirita olmaysiz. Agar siz \(b > 0\) ni eslab, logarifmni eksponensial shaklda yozsangiz, bu mantiqiy bo'ladi.
\[y = {\log _b}x\hspace{0.25in} \Rightarrow \hspace{0.25in}{b^y} = x\]\(b > 0\) beri \(x\) nol yoki manfiy bo'lishi mumkin emas. Shuning uchun siz logarifmaga manfiy son yoki nol qo'sha olmaysiz!
- \(f\left( x \right) = \ln \left( x \right)\) va \(g\left( x \right) = \log \left( x \right)\) grafigini chizing.
Yechim
Buni bir necha nuqtada baholash uchun kalkulyatordan foydalanish va keyin eskizni yaratishdan boshqa narsa emas. Mana eskiz.
Ushbu grafikdan biz har bir grafikning quyidagi xatti-harakatlarini ko'rishimiz mumkin.
\[\begin{align*} & \ln \left( x \right) \to \infty {\mbox{ as }}x \to \infty & \hspace{0.5in} & {\rm{and}} & \hspace{0.5in} & \ln \left( x \right) \to - \infty {\mbox{ as }}x \to 0\,\,\left( {x > 0} \right)\\ & \log \left( x \right) \to \infty {\mbox{ as }}x \to \infty & \hspace{0.5in} & {\rm{and}} & \hspace{0.5in} & \log \left( x \right) \to - \infty {\mbox{ as }}x \to 0\,\,\left( {x > 0} \right)\end{align*}\]Har bir logarifmda \(x > 0\) talab qilinishini unutmang.