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\(log_{a^4}b^4.log_ba^5=\dfrac{1}{4}.4.log_ab.5.log_ba=5.log_ab.log_ba=5\)
\(log_{a^3}b^2.log_ba^4=\dfrac{1}{3}.2.log_ab.4.log_ba=\dfrac{8}{3}.log_ab.log_ba=\dfrac{8}{3}\)
\(log_{a^{15}}b^7.log_{b^{49}}a^{30}=\dfrac{1}{15}.7.log_ab.\dfrac{1}{49}.30.log_ba=\dfrac{2}{7}log_ab.log_ba=\dfrac{2}{7}\)
\(log_{a^{2021}}b^{2020}.log_{b^{4040}}a^{6063}=\dfrac{1}{2021}.2020.log_ab.\dfrac{1}{4040}.6063.log_ba=\dfrac{3}{2}\)
\(log_{\sqrt{2}}\sqrt{2}=1;log_77=1\)
\(log_{10}1=0;log_91=0\)
\(3^{log_35}=5;7^{log_7\sqrt{2}}=\sqrt{2}\)
\(log_88^{-10}=-10;log_55^{\sqrt{3}}=\sqrt{3}\)
a.
ĐKXĐ: \(x>0\)
\(log_5x>6\Rightarrow x>6^5\Rightarrow x>7776\)
b.
ĐKXĐ: \(x>0\)
\(log_7x< 2\Rightarrow\left\{{}\begin{matrix}x>0\\x< 7^2\end{matrix}\right.\) \(\Rightarrow0< x< 49\)
c.
\(log_2x\le3\Rightarrow\left\{{}\begin{matrix}x>0\\x\le3^2\end{matrix}\right.\) \(\Rightarrow0< x\le9\)
d.
\(log_{\dfrac{1}{3}}x>27\Rightarrow\left\{{}\begin{matrix}x>0\\x< \left(\dfrac{1}{3}\right)^{27}\end{matrix}\right.\)
\(\Rightarrow0< x< \dfrac{1}{3^{27}}\)
\(log_216=log_22^4=4\)
\(log_32187=log_33^7=7\)
\(log_{10}\dfrac{1}{100}=log_{10}10^{-2}=-2\)
\(log10000=log10^4=4\)
\(9^{log_312}=3^{2log_312}=3^{log_3144}=144\)
\(8^{log_25}=2^{3log_25}=2^{log_2125}=125\)
\(\left(\dfrac{1}{25}\right)^{log_5\dfrac{1}{3}}=5^{-2log_5\dfrac{1}{3}}=5^{log_59}=9\)
\(\left(\dfrac{1}{4}\right)^{log_23}=2^{-2log_23}=2^{log_2\dfrac{1}{9}}=\dfrac{1}{9}\)
\(\left[log_24x\right]^2-log_{\sqrt{2}}2x=5\)
=>\(\left[log_2\left(2\cdot2x\right)\right]^2-log_{2^{\dfrac{1}{2}}}2x=5\)
=>\(\left[1+log_22x\right]^2-1:\dfrac{1}{2}\cdot log_22x=5\)
=>\(\left(log_22x\right)^2+2\cdot log_22x+1-2\cdot log_22x=5\)
=>\(\left(log_22x\right)^2=4\)
=>\(\left[{}\begin{matrix}log_22x=2\\log_22x=-2\left(loại\right)\end{matrix}\right.\Leftrightarrow log_22x=2\)
=>\(2x=2^2=4\)
=>x=2
\(log_5125=log_55^3=3\)
\(log_6216=log_66^3=3\)
\(log_{10}\dfrac{1}{10000}=log_{10}10^{-4}=-4\)
\(log\sqrt{1000}=log_{10}10^{\dfrac{3}{2}}=\dfrac{3}{2}\)
\(81^{log_35}=3^{3log_35}=3^{log_3125}=125\)
\(125^{log_52}=5^{3log_52}=5^{log_58}=8\)
\(\left(\dfrac{1}{49}\right)^{log_7\dfrac{1}{8}}=7^{-2log_7\dfrac{1}{8}}=7^{log_764}=64\)
\(\left(\dfrac{1}{625}\right)^{log_52}=5^{-4log_52}=5^{log_5\dfrac{1}{16}}=\dfrac{1}{16}\)
a: Bảng giá trị:
| x | 5 | 50 | 500 | 5000 |
| \(y=log2x\) | 1 | 2 | 3 | 4 |
Vẽ đồ thị:
b: bảng giá trị:
| x | 4 | 16 | 64 | 256 |
| \(y=log_{\dfrac{1}{4}}x\) | -1 | -2 | -3 | -4 |
Vẽ đồ thị:
1.
Ta có:
\(\left(n+1\right)^2=n^2+2n+1>n\left(n+2\right)\)
Lấy logarit 2 vế:
\(ln\left(n+1\right)^2>ln\left[n\left(n+2\right)\right]\)
\(\Rightarrow2ln\left(n+1\right)>ln\left(n\right)+ln\left(n+2\right)\ge2\sqrt{ln\left(n\right).ln\left(n+2\right)}\)
\(\Rightarrow ln^2\left(n+1\right)>ln\left(n\right).ln\left(n+2\right)\)
\(\Rightarrow\dfrac{ln\left(n+1\right)}{ln\left(n\right)}>\dfrac{ln\left(n+2\right)}{ln\left(n+1\right)}\)
\(\Rightarrow log_n\left(n+1\right)>log_{n+1}\left(n+2\right)\)
2.
\(\int\dfrac{x^3-1}{x^4+x}dx=\int\dfrac{2x^3-\left(x^3+1\right)}{x\left(x^3+1\right)}dx=\int\dfrac{2x^2}{x^3+1}dx-\int\dfrac{1}{x}dx\)
\(=\dfrac{2}{3}\int\dfrac{d\left(x^3+1\right)}{x^3+1}-\int\dfrac{dx}{x}\)
\(=\dfrac{2}{3}ln\left|x^3+1\right|-ln\left|x\right|+C\)
\(log_{a^3}b.log_ba=\dfrac{1}{3}.log_ab.log_ba=\dfrac{1}{3}\)
\(log_{a^{10}}b^5.log_{b^3}a^9=\dfrac{1}{10}.5.log_ab.\dfrac{1}{3}.9.log_ba=\dfrac{3}{2}\)
\(log_{a^{107}}b^{101}.log_{b^{303}}a^{428}=\dfrac{1}{107}.101.log_ab.\dfrac{1}{303}.428.log_ba=\dfrac{4}{3}.log_ab.log_ba=\dfrac{4}{3}\)
a: \(log_{a^3}b\cdot log_ba=\dfrac{1}{3}\cdot log_ab\cdot log_ba=\dfrac{1}{3}\)
b: \(log_{a^{10}}b^5\cdot log_{b^3}a^9\)
\(=\dfrac{1}{10}\cdot log_ab^5\cdot\dfrac{1}{3}\cdot log_ba^9\)
\(=\dfrac{1}{30}\cdot5\cdot log_ab\cdot9\cdot log_ba=\dfrac{45}{30}=\dfrac{3}{2}\)
c: \(log_{a^{107}}b^{101}\cdot log_{b^{303}}a^{428}\)
\(=\dfrac{1}{107}\cdot log_ab^{101}\cdot\dfrac{1}{303}\cdot log_ba^{428}\)
\(=\dfrac{1}{107}\cdot101\cdot log_ab\cdot\dfrac{1}{303}\cdot428\cdot log_ba\)
\(=4\cdot\dfrac{1}{3}=\dfrac{4}{3}\)