添加26届高一上学期第三、第四章剩余的知识梳理
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"K0410001X"
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],
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"content": "计算$\\sqrt{2}$的巴比伦算法所构造的递推公式是\\blank{90}."
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},
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"B00127": {
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"lesson": "K0202",
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"objs": [
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"K0202001B",
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"K0201004B",
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"K0202003B"
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],
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"content": "有理数指数幂:\\\\\n(1) 指数为零: $a^0=$\\blank{50}($a \\neq 0$);\\\\\n(2) 指数为负整数: $a^{-n}=$\\blank{50}($a \\neq 0$, $n$ 是正整数);\\\\\n(3) 指数为正分数: $a^{\\frac{m}{n}}=$\\blank{50}($a \\geq 0$, $m, n$是正整数, $n \\geq 2$, $(m, n)=1$);\\\\\n(4) 指数为负分数: $a^{-\\frac{m}{n}}=$\\blank{50}($a \\neq 0$, $m, n$是正整数, $n \\geq 2$, $(m, n)=1$)."
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},
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"B00128": {
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"lesson": "K0203",
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"objs": [
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"K0203001B",
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"K0203002B"
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],
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"content": "实数指数幂的性质(已知$a$\\blank{20}, $s,t\\in \\mathbf{R}$):\\\\\n(1) $a^s a^t=$\\blank{50};\\\\\n(2) $(a^s)^t=$\\blank{50};\\\\\n(3) $(ab)^t=$\\blank{50}."
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},
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"B00129": {
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"lesson": "K0203",
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"objs": [
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"K0203003B"
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],
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"content": "幂的基本不等式: 当$a>$\\blank{20}, $s>$\\blank{20}时, \\blank{20}$>$\\blank{20}."
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},
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"B00130": {
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"lesson": "K0204",
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"objs": [
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"K0204001B",
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"K0204002B"
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],
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"content": "对数的定义:\\\\\n(1) 在 $a>0$, $a \\neq 1$, 且 $N>0$ 的条件下, 唯一满足\\blank{50}的数 $x$, 称为 $N$ 以 $a$ 为底的对数, 并用符号\\blank{50}表示, 而 $N$ 称为\\blank{30}.\\\\\n(2) 从定义出发可以自然地得到一些常用的对数等式($a>0$, $a\\ne 1$, $N>0$, $b\\in \\mathbf{R}$):\\\\\n\\textcircled{1} $a^{\\log _a N}=$\\blank{30}; \\textcircled{2} $\\log_a a^b=$\\blank{30}; \\textcircled{3} $\\log _a 1=$\\blank{50}; \\textcircled{4} $\\log _a a=$\\blank{50}."
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},
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"B00131": {
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"lesson": "K0204",
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"objs": [
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"K0204003B"
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],
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"content": "常用对数与自然对数:\\\\\n(1) \\blank{100}称为常用对数, 记作\\blank{50};\\\\\n(2) 常数 $\\mathrm{e}$ 是\\blank{30}数, $e \\approx$\\blank{50}, \\blank{100}称为自然对数, 记作\\blank{50}."
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},
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"B00132": {
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"lesson": "K0205",
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"objs": [
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"K0205001B"
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],
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"content": "对数运算的基本性质($a>0$, $a\\ne 1$, $M,N\\in (0,+\\infty)$, $c\\in \\mathbf{R}$):\\\\\n(1) 对数性质 1: $\\log_a(MN)=$\\blank{100};\\\\\n(2) 对数性质 2: $\\log_a\\dfrac{M}{N}=$\\blank{100};\\\\\n(3) 对数性质 3: $\\log_aN^c=$\\blank{100}. 特别地, $\\log _a \\sqrt[n]{M}=$\\blank{50}($n$ 为大于 $1$ 的正整数)."
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},
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"B00133": {
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"lesson": "K0206",
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"objs": [
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"K0206001B",
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"K0206003B"
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],
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"content": "对数换底公式(已知$a>0$, $a\\ne 1$, $b>0$, $b\\ne 1$, $N>0$): $\\log_a N=\\dfrac{\\ \\blank{50}\\ }{\\ \\blank{50}\\ }$.\\\\\n推论1: $\\log _a b\\cdot$\\blank{50}$=1$;\\\\\n推论2: $\\log_{a^m}N^n=$\\blank{50}($m,n\\in \\mathbf{R}$)."
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},
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"B00134": {
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"lesson": "K0208",
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"objs": [
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"K0208002B"
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],
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"content": "当幂函数的指数为正数时, 它在 $[0,+\\infty)$上是\\blank{50}(单调性); 当幂函数的指数为负数时, 它在 ($0,+\\infty$)上是\\blank{50}.(单调性)"
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},
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"B00135": {
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"lesson": "K0208",
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"objs": [
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"K0208003B"
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],
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"content": "幂函数的图像必过定点\\blank{50}."
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},
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"B00136": {
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"lesson": "K0208",
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"objs": [
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"K0208005B"
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],
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"content": "通过图像的平移可以直观地分析与幂函数密切相关的函数的一些性质. 例如: 函数 $y=\\dfrac{2 x+7}{x+3}$ 的图像可视为函数 $y=\\dfrac{1}{x}$ 的图像按\\blank{200}平移所得, 因此它的单调减区间为\\blank{100}, 值域为\\blank{100}."
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},
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"B00137": {
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"lesson": "K0209",
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"objs": [
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"K0209001B"
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],
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"content": "当底数 $a$ 固定, 且\\blank{100}时, 等式 $y=a^x$ 确定了变量 $y$ 随变量变化的规律, 称为底为 $a$ 的指数函数."
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},
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"B00138": {
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"lesson": "K0209",
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"objs": [
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"K0209002B"
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],
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"content": "指数函数的定义域为\\blank{50}."
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},
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"B00139": {
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"lesson": "K0210",
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"objs": [
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"K0210001B",
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"K0210002B",
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"K0210005B"
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],
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"content": "函数的图像与性质密切相关, 对于指数函数而言:\n\\begin{center}\n\\begin{tabular}{|c|>{\\centering\\arraybackslash}p{5cm}|>{\\centering\\arraybackslash}p{5cm}|}\n\\hline\n$y=a^x$ & $a>1$ & $0<a<1$\\\\\n\\hline\n图像 & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,0.5) node [above] {$y=$\\blank{10}}-- (2,0.5);\n\\end{tikzpicture} & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,0.5) node [above] {$y=$\\blank{10}}-- (2,0.5);\n\\end{tikzpicture} \\\\\n\\hline\n\\multirow{3}{*}{图像特征} & \\multicolumn{2}{l|}{\\textcircled{1} 图像都在$x$轴\\blank{30}, \\blank{80}于$x$轴, 但永不\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 过点\\blank{50}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} \\\\ \\hline\n\\multirow{3}{*}{函数性质} & \\multicolumn{2}{l|}{\\textcircled{1} 定义域为\\blank{50}, 函数值恒\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 当$x=$\\blank{30}时, $y=$\\blank{30}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 在$\\mathbf{R}$上是\\blank{50}函数.} & \\multicolumn{1}{l|}{\\textcircled{3} 在$\\mathbf{R}$上是\\blank{50}函数.} \\\\ \\hline\n\\end{tabular}\n\\end{center}"
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},
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"B00140": {
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"lesson": "K0211",
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"objs": [
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"K0211002B"
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],
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"content": "如图表示几个地区的某物种关于时间的图形, 找出与下面每个描述相符的图形并描述剩余图形.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (4,0) node [right] {时间(年)};\n\\draw [->] (0,0) -- (0,4) node [above] {物种数量};\n\\draw [domain = 0:{ln(4)/ln(1.08)/8}] plot (\\x,{exp(8*\\x*ln(1.08))}) node [above] {\\textcircled{1}};\n\\draw [domain = 0:{ln(4)/ln(1.05)/8}] plot (\\x,{exp(8*\\x*ln(1.05))}) node [above] {\\textcircled{2}};\n\\draw (0,1.2) -- (4,2.5) node [right] {\\textcircled{3}};\n\\draw [domain = 0:4] plot (\\x,{3.5*exp(8*\\x*ln(0.95))}) node [right] {\\textcircled{4}};\n\\draw (0,1.8) -- (4,1.8) node [right] {\\textcircled{5}};\n\\draw (0,2.3) -- (4,1.3) node [right] {\\textcircled{6}};\n\\end{tikzpicture}\n\\end{center}\n(1) 物种数量每年增长 $5 \\%$: \\blank{30};\\\\\n(2) 物种数量每年增长 $8 \\%$: \\blank{30};\\\\\n(3) 物种数量每年增加 $5000$ 人: \\blank{30};\\\\\n(4) 物种数量保持不变的: \\blank{30};\\\\\n剩余两个图形分别可以描述为:\\blank{150}, \\blank{150}."
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},
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"B00141": {
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"lesson": "K0212",
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"objs": [
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"K0212001B"
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],
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"content": "当底数 $a$ 固定, 且\\blank{80}时, $x$ 以 $a$ 为底的对数确定了变量 $y$ 随变量\\blank{30}变化的规律, 称为底为 $a$ 的对数函数."
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},
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"B00142": {
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"lesson": "K0212",
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"objs": [
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"K0212002B"
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],
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"content": "对数函数的定义域为\\blank{50}."
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},
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"B00143": {
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"lesson": "K0213",
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"objs": [
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"K0213005B",
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"K0213006B"
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],
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"content": "对数函数与同底的指数函数互为\\blank{50}, 它们的图像间的关系是\\blank{100}."
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},
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"B00144": {
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"lesson": "K0213",
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"objs": [
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"K0213007B"
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],
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"content": "函数的图像与性质密切相关, 对于对数函数而言:\n\\begin{center}\n\\begin{tabular}{|c|>{\\centering\\arraybackslash}p{5cm}|>{\\centering\\arraybackslash}p{5cm}|}\n\\hline\n$y=\\log_a x$ & $a>1$ & $0<a<1$\\\\\n\\hline\n图像 & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0.5,0) node [below] {$1$};\n\\end{tikzpicture} & \\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0.5,0) node [below] {$1$};\n\\end{tikzpicture} \\\\\n\\hline\n\\multirow{3}{*}{图像特征} & \\multicolumn{2}{l|}{\\textcircled{1} 图像都在$y$轴\\blank{30}, \\blank{80}于$y$轴, 但永不\\blank{30}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 过点\\blank{50}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} & \\multicolumn{1}{l|}{\\textcircled{3} 由左至右图像\\blank{30}.} \\\\ \\hline\n\\multirow{3}{*}{函数性质} & \\multicolumn{2}{l|}{\\textcircled{1} 定义域为\\blank{50}.}\\\\ \n\\cline{2-3} & \\multicolumn{2}{l|}{\\textcircled{2} 当$x=$\\blank{30}时, $y=$\\blank{30}.} \\\\ \n\\cline{2-3} & \\multicolumn{1}{l|}{\\textcircled{3} 在区间\\blank{30}上是\\blank{30}函数.} & \\multicolumn{1}{l|}{\\textcircled{3} 在区间\\blank{30}上是\\blank{30}函数.} \\\\ \\hline\n\\end{tabular}\n\\end{center}"
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},
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"B00145": {
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"lesson": "K0214",
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"objs": [
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"K0214002B"
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],
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"content": "当$a>1$, $b\\in \\mathbf{R}$时, 关于$x$的不等式$\\log_a x>b$的解集为\\blank{50}.\\\\\n这是因为当$\\log_a$\\blank{50}$=b$. 当$x\\in $\\blank{50}时, 因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x>b$, 这表明区间\\blank{50}中的实数$x$都是解; 而当$x\\in $\\blank{50}时, 同样因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x\\le b$, 这表明区间\\blank{50}中的实数$x$都不是解."
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},
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"B00146": {
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"lesson": "K0214",
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"objs": [
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"K0214002B"
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],
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"content": "当$0<a<1$, $b\\in \\mathbf{R}$时, 关于$x$的不等式$\\log_a x>b$的解集为\\blank{50}.\\\\\n这是因为当$\\log_a$\\blank{50}$=b$. 当$x\\in $\\blank{50}时, 因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x>b$, 这表明区间\\blank{50}中的实数$x$都是解; 而当$x\\in $\\blank{50}时, 同样因为函数$y=\\log_a x$是\\blank{50}函数, 所以$\\log_a x\\le b$, 这表明区间\\blank{50}中的实数$x$都不是解."
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}
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}
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