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求解指数方程示例

求解指数方程示例
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2015-05-15 02:58
解析几何函数图像
本范例演示如何使用 GeoGebra 求解指数方程 10^(x-3) = 208，并展示求解步骤。涉及的核心数学概念包括指数方程和常用对数。作图要点使用了 LaTeX 命令、Corner 对象、stepMax、stepper、steps、counter、RandomizationText、Original、Randomized、A、B、xmin、xmax、ymin、ymax、button1、base、subtract、constant、text9、Problem、text1、text4、text5、text6、ans、finalans、text10、text11、text2_1、text3_1、text2_2、text3_2、text3、text2_3、text3_3、ans_1、text2_5、text3_4、text8、text0、text12 等GeoGebra命令和对象。适合初中及以上数学教学和自学使用。

GeoGebra 命令及参数

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  3. Corner(1) → A
  4. Corner(3) → B
  5. LaTeX((("\text{Add }\textcolor{#000099}{" * subtract) * "}\text{ to both sides.}")) → text9
  6. LaTeX((((((("\text{Solve for }\textcolor{#000099}{x}\text{ in the equation }\textcolor{#000099}{" * base) * "^{x-") * subtract) * "}=") * constant) * "}.")) → Problem
  7. LaTeX("\text{Notice that }\textcolor{#000099}{x}\text{ is in the exponent. In order to extract }\textcolor{#000099}{x}\text{ from the}\cr\text{argument of any function, apply the inverse of the function to both sides of the}\cr\text{equation.") → text1
  8. LaTeX((((((("\textcolor{#000099}{" * base) * "^{x-") * subtract) * "}=") * constant) * "}")) → text4
  9. LaTeX((("\text{Take the logarithm base }\textcolor{#000099}{" * base) * "}\text{ both sides.}")) → text5
  10. LaTeX((((((((((("\textcolor{#000099}{log_{" * base) * "}\left(") * base) * "^{x-") * subtract) * "}\right)=log_{") * base) * "}(") * constant) * ")}")) → text6
  11. LaTeX((("\textcolor{#000099}{x\approx" * finalans) * "}")) → text10
  12. LaTeX((((((((("\text{The value of }\textcolor{#000099}{x}\text{ in the exponential equation }\textcolor{#000099}{" * base) * "^{x-") * subtract) * "}=") * constant) * "}\text{ is approximately }\textcolor{#000099}{") * finalans) * "}.")) → text11
  13. LaTeX("\textcolor{#000099}{y=a^x}") → text2_1
  14. LaTeX("\text{General form of an exponential equation}") → text3_1
  15. LaTeX("\textcolor{#000099}{\log_{a}(y)=x}") → text2_2
  16. LaTeX("\text{Rewrite in logarithmic form.}") → text3_2
  17. LaTeX("\text{Given exponential equation}") → text3
  18. LaTeX((((((("\textcolor{#000099}{\log_{(" * base) * ")}(") * constant) * ")=(x-") * subtract) * ")}")) → text2_3
  19. LaTeX("\text{Substitute.}") → text3_3
  20. LaTeX((((((("\textcolor{#000099}{\log_{(" * base) * ")}(") * constant) * ")=(x-") * subtract) * ")}")) → text2_5
  21. LaTeX("\text{Logarithmic form of the original equation}") → text3_4
  22. LaTeX((((("\textcolor{#000099}{(" * ans) * ")\approx x-") * subtract) * "}")) → text8
  23. LaTeX("\text{Move the slider to solve the problem.}") → text0
  24. LaTeX("\text{An exponential equation of the form }\textcolor{#000099}{y=a^x}\text{ can be rewritten }\cr\text{as a logarithmic equation of the form }\textcolor{#000099}{\log_{a}(y)=x}.") → text2
  25. LaTeX((((((((((((("\text{The exponential equation }\textcolor{#000099}{" * base) * "^{x-") * subtract) * "}=") * constant) * "}\text{ can be rewritten as the }\cr\text{logarithmic equation }\textcolor{#000099}{\log_{(") * base) * ")}(") * constant) * ")=(x-") * subtract) * ")}.")) → text2_4
  26. LaTeX("\text{Find the common logarithm with}\cr\text{a calculator and substitute.}") → text7

AI 绘图提示语参考

绘制一个展示如何解指数方程 10^(x-3) = 208 的图形，包含步骤说明和关键数学概念。图形应显示指数方程和对应的对数形式，以及解题步骤的文字说明。首先展示原始方程，然后逐步展示解题过程，包括将对数应用于等式两边，最后显示解的近似值。图形应包含指数函数 y=a^x 和对数函数 log_a(y)=x 的示意图，以及它们之间的关系说明。
以上提示语为根据原始绘图命令整理而来，仅供参考。