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CCGPS AA 4.2.3 Example 3: Simplifying a Logarithmic Function Using Log Propertie

CCGPS AA 4.2.3 Example 3: Simplifying a Logarithmic Function Using Log Propertie
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2014-06-20 04:58
FunctionSequenceSlider Animation
In collections CCGPS AA Unit 4 Lesson 2
① This example interactively demonstrates how to simplify the logarithmic function f(x)=2 log_x – log_(x–3) by applying logarithm properties step by step. ② Core concepts include the product, quotient, and power rules of logarithms, domain considerations, and algebraic manipulation of log expressions. ③ Construction highlights: GeoGebra commands Sequence, If, LaTeX, Text, TableText; objects such as sliders (step, stepmax, stepper), logpropertywords, addsubtractwords, logquot, logprod, logprop, logdivmult, simpdivmult, and button controls guide users through the simplification process. ④ Designed for high‑school algebra students (grades 10‑12) and teachers creating interactive lessons on logarithmic functions.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create an interactive graphic demonstrating the simplification of the logarithmic function f(x)=2 log_x – log_(x–3) using logarithmic properties step by step, featuring a slider to adjust parameters, with annotations highlighting key steps, applied properties like the quotient and product rules, relevant mathematical formulas, and explanatory text.
The prompt above was compiled from the original drawing commands and is for reference only.