3.2-bo'lim: Asosiy logarifmik funksiyalar

1. Kalkulyatorsiz quyidagi logarifmlarning har birining aniq qiymatini bering.
   1. \({\log _2}16\)
   2. \({\log _4}16\)
   3. \({\log _5}625\)
   1. \(\displaystyle {\log _9}\frac{1}{{531441}}\)
   2. \(\displaystyle {\log _{\frac{1}{6}}}36\)
   3. \(\displaystyle {\log _{\frac{3}{2}}}\frac{{27}}{8}\)
   Yechim

   Bularni kalkulyatorsiz bajarish uchun quyidagilarni eslab qolishingiz kerak.

   \[y = {\log _b}x\hspace{0.25in}\hspace{0.25in}{\mbox{is equivalent to }}\hspace{0.25in}\hspace{0.25in}x = {b^y}\]

   Bu erda \(b\), \(b > 0\) va \(b \ne 1\) kabi har qanday raqam asos deb ataladi. Birinchisi odatda chaqiriladi logarifmik shakl ikkinchisi esa odatda deyiladi eksponensial shakl. Logarifmik shakl “\(y\), \(x\) ning \(b\) log bazasiga teng” deb o'qiladi.

   Shunday qilib, agar siz logarifmlarni eksponensial shaklga aylantirsangiz, odatda bunday logarifmlarni hisoblash juda oson.

   (a) \({\log _2}16 = 4\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{2^4} = 16\)

   (b) \({\log _4}16 = 2\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{4^2} = 16\)

   O'rtasidagi farqga e'tibor bering (a) va (b)! Logarifmda foydalanadigan \(b\) bazasi juda muhim! Boshqa baza, deyarli har bir holatda, boshqacha javob beradi. Siz har doim bazaga e'tibor berishingiz kerak!

   (c) \({\log _5}625 = 4\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{5^4} = 625\)

   (d) \(\displaystyle {\log _9}\frac{1}{{531441}} = - 6\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{9^{ - 6}} = \frac{1}{{{9^6}}} = \frac{1}{{531441}}\)

   (e) \(\displaystyle {\log _{\frac{1}{6}}}36 = - 2\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{\left( {\frac{1}{6}} \right)^{ - 2}} = {6^2} = 36\)

   (f) \(\displaystyle {\log _{\frac{3}{2}}}\frac{{27}}{8} = 3\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{\left( {\frac{3}{2}} \right)^3} = \frac{{27}}{8}\)

2. Kalkulyatorsiz quyidagi logarifmlarning har birining aniq qiymatini bering.
   1. \(\ln \sqrt[3]{{\bf{e}}}\)
   2. \(\log 1000\)
   3. \({\log _{16}}16\)
   1. \({\log _{23}}1\)
   2. \({\log _2}\sqrt[7]{{32}}\)
   Yechim

   Avval hal qilinishi kerak bo'lgan bir nechta tezkor notatsiya muammolari mavjud.

   \[\begin{align*}\ln x & = {\log _{\bf{e}}}x & \hspace{0.5in} & {\mbox{This log is called the natural logarithm}}\\ \log x & = {\log _{10}}x & \hspace{0.5in} & {\mbox{This log is called the common logarithm}}\end{align*}\]

   Natural logarifmdagi \(e\) yuqoridagi 2-masalada ishlatilgan \(e\) bilan bir xil. Umumiy logarifm va natural logarifm logarifmlar boshqa logarifmalarga qaraganda tez-tez uchraydi, shuning uchun maxsus belgi va maxsus nomlarga o'rganing.

   Ushbu to'plamdagi logarifmlarni baholash uchun talab qilinadigan ish avvalgi masaladagi masala bilan bir xil.

   (a) \(\ln \sqrt[3]{{\bf{e}}} = \frac{1}{3}\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{{\bf{e}}^{\frac{1}{3}}} = \sqrt[3]{{\bf{e}}}\)

   (b) \(\log 1000 = 3\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{10^3} = 1000\)

   (c) \({\log _{16}}16 = 1\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{16^1} = 16\)

   (d) \({\log _{23}}1 = 0\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}{23^0} = 1\)

   (e) \({\log _2}\sqrt[7]{{32}} = \frac{5}{7}\hspace{0.25in}\hspace{0.25in}{\rm{because}}\hspace{0.25in}\hspace{0.25in}\sqrt[7]{{32}} = {32^{\frac{1}{7}}} = {\left( {{2^5}} \right)^{\frac{1}{7}}} = {2^{\frac{5}{7}}}\)