Logarithmic Equation Solving Example: Solve log_5(x-20)=2

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2015-05-15 02:56

Function Geometric Construction

This example demonstrates solving the logarithmic equation log_5(x-20)=2, involving concepts of logarithmic functions, exponential relationships, and variable solving. Construction uses GeoGebra commands like LaTeX, Corner, stepMax, stepper, steps, counter, etc. Suitable for high school students learning logarithmic function properties and equation solving, useful for math teaching and self-study.

GeoGebra Commands & Parameters

LaTeX(steps) → counter
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
Corner(1) → A
Corner(3) → B
LaTeX((((((("\textcolor{#000099}{log_{" * base) * "}(x-") * subtract) * ")=") * constant) * "}")) → text4
LaTeX("\text{Simplify the left side of the equation using the fact that }\textcolor{#000099}{a^{\log_{a}(x)}=x}\text{. Simplify the}\cr\text{power on the right side of the equation.}") → text7
LaTeX((("\textcolor{#000099}{x=" * Simplify[base^(constant) + subtract]) * "}")) → text10
LaTeX((((((((("\text{The value of }\textcolor{#000099}{x}\text{ in the logarithmic equation }\textcolor{#000099}{\log_{" * base) * "}(x-") * subtract) * ")=") * constant) * "}\text{ is }\textcolor{#000099}{") * Simplify[base^(constant) + subtract]) * "}.")) → text11
LaTeX((((((("\text{Solve for }\textcolor{#000099}{x}\text{ in the equation }\textcolor{#000099}{log_{" * base) * "}(x-") * subtract) * ")=") * constant) * "}.")) → Problem
LaTeX("\text{Notice that }\textcolor{#000099}{x}\text{ is in the argument of the logarithm. In order to extract }\textcolor{#000099}{x}\text{ from the argument of any function, }\cr\text{apply the inverse of the function to both sides of the equation.") → text1
LaTeX("\text{Given logarithmic equation}") → text3
LaTeX("\textcolor{#000099}{y=\log_{a}(x)}") → text2_1
LaTeX("\textcolor{#000099}{a^y=x}") → text2_2
LaTeX((((((("\textcolor{#000099}{(" * base) * ")^{(") * constant) * ")}=(x-") * subtract) * ")}")) → text6
LaTeX("\text{Solve the result for x.}") → text5_1
LaTeX("\text{Rewrite in exponential form.}") → text3_1
LaTeX((((((("\textcolor{#000099}{(" * base) * ")^{(") * constant) * ")}=(x-") * subtract) * ")}")) → text6_1
LaTeX("\text{Exponential form of the original equation}") → text5_2
LaTeX((((("\textcolor{#000099}{" * Simplify[base^(constant)]) * "=x-") * subtract) * "}")) → text8
LaTeX("\text{Rewrite the equation in exponential form.}") → text5_3
LaTeX("\text{Simplify.}") → text5
LaTeX("\text{Move the slider to solve the problem.}") → text0
LaTeX("\text{General form of a logarithmic equation}") → text3_2
LaTeX("\text{A logarithmic equation of the form }\textcolor{#000099}{y=\log_{a}(x)}\text{ can be rewritten }\cr\text{as an exponential equation of the form }\textcolor{#000099}{a^y=x}.") → text2
LaTeX((((((((((((("\text{The logarithmic equation }\textcolor{#000099}{log_{" * base) * "}(x-") * subtract) * ")=") * constant) * "}\text{ can be rewritten}\cr\text{as an exponential equation of the form }\textcolor{#000099}{(") * base) * ")^{(") * constant) * ")}=(x-") * subtract) * ").}")) → text2_3

AI Drawing Prompt Reference

Create a visual example demonstrating the solution of the logarithmic equation log_5(x-20)=2. Include the graph of the logarithmic function, the transformation between logarithmic and exponential relationships, and the step-by-step derivation from the logarithmic form to the exponential form to solve for x. Show the original equation, then explain how to simplify and solve using the property a^y=x, leading to the final value of x. The graphic should illustrate the properties of logarithmic functions and the steps in solving equations.

The prompt above was compiled from the original drawing commands and is for reference only.