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Solving an Exponential Equation Example by Walch Education

Solving an Exponential Equation Example by Walch Education
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2015-05-15 02:58
Coordinate GeometryGraphing
This example demonstrates how to solve the exponential equation 10^(x-3) = 208 using GeoGebra, including step-by-step solutions. Key math concepts involved are exponential equations and logarithms. GeoGebra commands and objects used in construction include LaTeX, Corner, 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. Suitable for middle school and above mathematics teaching and self-learning.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create a visual representation of solving the exponential equation 10^(x-3) = 208, including step-by-step solutions and key mathematical concepts. The graphic should display the exponential equation, its logarithmic form, and textual explanations of the solution steps. Start with the original equation, then show the step-by-step process of applying logarithms to both sides, and conclude with the approximate solution. Include illustrations of the exponential function y=a^x and the logarithmic function log_a(y)=x, along with explanations of their relationship.
The prompt above was compiled from the original drawing commands and is for reference only.