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CCGPS AA 4.2.3 示例 3：利用对数性质化简对数函数

CCGPS AA 4.2.3 示例 3：利用对数性质化简对数函数
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2014-06-20 04:58
函数数列滑块动画
所属专题 CCGPS AA Unit 4 Lesson 2
① 本例通过交互式步骤展示如何利用对数性质化简对数函数 f(x)=2 log_x – log_(x–3)。② 核心数学概念包括对数乘法、除法、幂则，定义域分析及代数化简。③ 构造亮点：GeoGebra 命令 Sequence、If、LaTeX、Text、TableText；对象包括步进滑块 (step、stepmax、stepper)、logpropertywords、addsubtractwords、logquot、logprod、logprop、logdivmult、simpdivmult，以及按钮控件，引导用户逐步应用性质进行化简。④ 适合高中 algebra 学生（10‑12 年级）及教师，用于讲授或演示对数运算的交互式教学。

GeoGebra 命令及参数

  1. Sequence(x = a, a, -7, -5, 0.5) → align
  2. If(const2 < 0, "-", "+") → pm
  3. LaTeX((((("\text{x} {" * pm) * "} {") * abs(const2)) * "}")) → expression
  4. LaTeX((((("\textcolor{#000099}{\text{f(x)=}{" * e) * "}\log{x}-\log{\left(") * expression) * "\right)}}")) → divtext
  5. LaTeX((((("\textcolor{#000099}{\text{f(x)=}{" * e) * "}\log{x}+\log{\left(") * expression) * "\right)}}")) → multtext
  6. If(divormult ≟ 1, divtext, multtext) → divmult
  7. LaTeX("\text{Identify which logarithmic properties can be used to simplify the}\cr \text{logarithmic function.}") → text2
  8. If(divormult ≟ 1, LaTeX["\textcolor{#000099}\text{Quotient Property of Logarithms}"], LaTeX["\textcolor{#000099}\text{Product Property of Logarithms}"]) → logpropertywords
  9. If(divormult ≟ 1, "subtraction", "addition") → addsubtractwords
  10. LaTeX("\textcolor{#000099}{\log_{a}{\left(\frac{x}{y} \right)}=\log_{a}{x} - \log_{a}{y}}") → logquot
  11. LaTeX("\textcolor{#000099}{\log_{a}{\left(xy \right)}=\log_{a}{x} + \log_{a}{y}}") → logprod
  12. If(divormult ≟ 1, logquot, logprod) → logprop
  13. LaTeX((((((("\text{Additionally, this function includes the }{" * addsubtractwords) * "}\text{ of two logarithmic}\cr \text{terms. This form is also seen in the }{") * logpropertywords) * "}\text{,}\cr\text{where }{") * logprop) * "}\text{.}")) → text4
  14. LaTeX("\text{Simplify the logarithmic function by applying the identified properties.}") → text6
  15. If(divormult ≟ 1, LaTeX[(((("\textcolor{#000099}{f(x)=\left(\log{x^{" * e) * "}}\right) - \log \left({") * expression) * "}\right) }")], LaTeX[(((("\textcolor{#000099}{f(x)=\left(\log{x^{" * e) * "}}\right) + \log \left({") * expression) * "}\right) }")]) → logdivmult
  16. If(divormult ≟ 1, LaTeX[(((("\textcolor{#000099}{f(x)=\log \left(\frac{x^{" * e) * "}}{{") * expression) * "}}\right)}")], LaTeX[(((("\textcolor{#000099}{f(x)=\log \left(x^{" * e) * "}\left({") * expression) * "}\right)\right)}")]) → simpdivmult
  17. LaTeX((((("\text{The logarithmic function }{" * divmult) * "}\cr \text{simplifies to }{") * simpdivmult) * "}\text{.}")) → text12
  18. LaTeX((("\textcolor{#000099}{" * divmult) * "}")) → text1
  19. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text13
  20. Text("Move the slider to solve the problem.") → text0
  21. LaTeX((("\text{Both the }\textcolor{#000099}\text{Power Property of Logarithms}\text{ and the }{" * logpropertywords) * "}\cr \text{can be used to simplify the given function.}")) → text5
  22. LaTeX((((("\text{Thus, }\textcolor{#000099}{{" * e) * "}\, \log{x}=\log{x^{") * e) * "}}}\text{.}")) → text9
  23. LaTeX((("\text{Apply the }\textcolor{#000099}\text{Power Property of Logarithms}\text{ to the term }\textcolor{#000099}{{" * e) * "}\, \log {x}}\text{ from the}\cr\text{original function. The base, }\textcolor{#000099}\text{a}\text{, is understood to be }\textcolor{#000099}\text{10}\text{ and is omitted.}")) → text7
  24. LaTeX((((("\text{The given function contains the term }\textcolor{#000099}{{" * e) * "}\, \log{x}}\text{. This is the same as}\cr \textcolor{#000099}{") * e) * " \cdot \log{x}}\text{. This form is also seen in the Power Property of Logarithms,}\cr \text{where }\textcolor{#000099}{\log_{a}{x^{y}} = y \cdot \log_{a}{x}}\text{.}")) → text3
  25. TableText({{LaTeX["\textcolor{#000099}{y\, \log{x}=\log{x^y}}"], LaTeX["\text{Power Property of Logarithms}"]}, {LaTeX[(((("\textcolor{#000099}{{\left(" * e) * "\right)}\, \log{x} = \log{x^{\left(") * e) * "\right)}}}")], LaTeX[(("\text{Substitute }\textcolor{#000099}{" * e) * "}\text{ for y.}")]}}, "l") → text8
  26. LaTeX((((((((("\text{Now apply the }{" * logpropertywords) * "}\text{, }{") * logprop) * "}\text{,}\cr \text{to the original function. Use }\textcolor{#000099}{\log{x^{") * e) * "}}}\text{ in place of }\textcolor{#000099}{{") * e) * "}\, \log{x}}\text{.}")) → text10
  27. TableText({{LaTeX[(("{" * divmult) * "}")], LaTeX["\text{Original function}"]}, {LaTeX[(("{" * logdivmult) * "}")], LaTeX[(((("\text{Substitute }\textcolor{#000099}{\log{x^{" * e) * "}}}\text{ for }\textcolor{#000099}{{") * e) * "}\, \log{x}}\text{.}")]}, {LaTeX[(("{" * simpdivmult) * "}")], LaTeX[(("\text{Simplify using the }{" * logpropertywords) * "}\text{.}")]}}, "l") → text11

AI 绘图提示语参考

绘制一个演示对数函数简化的交互式图形，包含一个滑动条用于调整参数，展示如何通过对数性质逐步简化函数 f(x)=2 log_x – log_(x–3)，并标注关键步骤和使用的对数性质，如商数性质和乘法性质，以及相关的数学公式和文字说明。
以上提示语为根据原始绘图命令整理而来，仅供参考。