MathPlotAI Logo MathPlotAI

CCGPS AA 4.2.3 Example 2: Demonstrating the Product Property of Logarithms to Pr

CCGPS AA 4.2.3 Example 2: Demonstrating the Product Property of Logarithms to Pr
Loading the math board and drawing, please wait…
2014-06-20 04:53
FunctionSequenceSlider Animation
In collections CCGPS AA Unit 4 Lesson 2
This GeoGebra example dynamically illustrates the product property of logarithms, proving that log_a(xy) = log_a x + log_a y. It covers the definition of logarithms, the product property, and its algebraic proof. The construction uses Sequence, Text, LaTeX, and TableText commands together with step sliders (stepmax, stepper, steps) and multiple text objects (productproperty, mtext, ntext, etc.) to generate and update expressions step by step. It is suitable for high‑school algebra or pre‑calculus courses, helping students visualize logarithmic operations.

GeoGebra Commands & Parameters

  1. Sequence(x = a, a, -7, -5, 0.5) → align
  2. Text("Move the slider to solve the problem.") → text0
  3. LaTeX("\textcolor{#000099}{\log_{a}{\left(xy\right)}=\log_{a}{x}+\log_{a}{y}}") → productproperty
  4. LaTeX("\textcolor{#000099}{M=\log_{a}{x}}") → mtext
  5. LaTeX("\textcolor{#000099}{N=\log_{a}{y}}") → ntext
  6. LaTeX("\text{Write the equivalent exponential equation for }\textcolor{#000099}{\log_{a}{x}}\text{.} ") → text2
  7. LaTeX((((("\text{To write the equivalent exponential equation, let }{" * mtext) * "}\text{.} \cr \text{By the definition of logarithms, }{") * mtext) * "}\text{ is equivalent to }\textcolor{#000099}{a^{x} = M}\text{.}")) → text3
  8. LaTeX("\text{Write the equivalent exponential equation for }\textcolor{#000099}{\log_{a}{y}}\text{.} ") → text4
  9. LaTeX((((("\text{To write the equivalent exponential equation, let }{" * ntext) * "}\text{.} \cr \text{By the definition of logarithms, }{") * ntext) * "}\text{ is equivalent to }\textcolor{#000099}{a^{y} = N}\text{.}")) → text5
  10. LaTeX("\text{Write the exponential form of the Product Property.}") → text6
  11. LaTeX("\text{According to the Product Property, }\textcolor{#000099}{\log_{a}{ \left(xy\right)} = M + N}\text{.} \cr \text{Therefore, the exponential form of the Product Property is }\textcolor{#000099}{a^{\left(M+N\right)} = xy}\text{.} \cr \text{By the Product Property of Exponents, }\textcolor{#000099}{a^{\left(M+N\right)} = \left(a^{M}\right)\left(a^{N}\right)}\text{.}") → text7
  12. LaTeX("\text{Substitute specific values for }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{x}\text{, and }\textcolor{#000099}{y}\text{ into the original equation}\cr \text{to verify that the Product Property of Logarithms holds true for}\cr \text{this problem.}") → text8
  13. LaTeX((("\text{The original equation was }{" * productproperty) * "}\text{.}")) → text9
  14. LaTeX("\text{Start by substituting values into the left-hand side of the equation, }\cr \textcolor{#000099}{\log_{a}{ \left(xy\right)}}\text{, and evaluating. Then, substitute the same values into}\cr
\textcolor{#000099}{\log_{a}{x} + \log_{a}{y}}\text{ and solve to see if the result matches that for }\textcolor{#000099}{\log_{a}{ \left(xy\right)}}\text{.}") → text10
  15. LaTeX("\text{For example, let }\textcolor{#000099}{a=10}\text{, }\textcolor{#000099}{x=3}\text{, and }\textcolor{#000099}{y=5}\text{.}") → text11
  16. LaTeX("\text{Now evaluate the terms in the right-hand side of the equation, }\cr \textcolor{#000099}{\log_{a}{x} + \log_{a}{y}}\text{, for the same values of }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{x}\text{, and }\textcolor{#000099}{y}\text{.}") → text13
  17. LaTeX("\text{The results match: }\textcolor{#000099}{\log_{10}{15}\approx{ 1.18}}\text{ and }\textcolor{#000099}{\log_{10}{3} + \log_{10}{5} \approx {1.18}}\text{.}\cr \text{Therefore, the Product Property of Logarithms holds true.}") → text15
  18. TableText({{LaTeX["\textcolor{#000099}{\log_{a}{ \left(xy\right)}}"], LaTeX["\text{Original expression}"]}, {LaTeX["\textcolor{#000099}{= \log_{\left(10\right)}{\left[(3)(5)\right]}}"], LaTeX["\text{Substitute }\textcolor{#000099}{10}\text{ for }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{3}\text{ for }\textcolor{#000099}{x}\text{, and }\textcolor{#000099}{5}\text{ for }\textcolor{#000099}{y}\text{.}"]}, {LaTeX["\textcolor{#000099}{= \log_{10}{15}}"], LaTeX["\text{Multiply the numbers.}"]}, {LaTeX["\textcolor{#000099}{\approx {1.18}}"], LaTeX["\text{Use a calculator to compute }\textcolor{#000099}{\log_{10}{15}}\text{.}"]}}, "l") → text12
  19. TableText({{LaTeX["\textcolor{#000099}{\log_{a}{x}+\log_{a}{y}}"], LaTeX["\text{Original expression}"]}, {LaTeX["\textcolor{#000099}{= \log_{\left(10\right)}{\left(3\right)}+\log_{\left(10\right)}{\left(5\right)}}"], LaTeX["\text{Substitute }\textcolor{#000099}{10}\text{ for }\textcolor{#000099}{a}\text{, }\textcolor{#000099}{3}\text{ for }\textcolor{#000099}{x}\text{, and }\textcolor{#000099}{5}\text{ for }\textcolor{#000099}{y}\text{.}"]}, {LaTeX["\textcolor{#000099}{\approx 0.48 + 0.70}"], LaTeX["\text{Use a calculator to compute }\textcolor{#000099}{\log_{10}{3}}\text{ and }\textcolor{#000099}{\log_{10}{5}}\text{.}"]}, {LaTeX["\textcolor{#000099}{\approx {1.18}}"], LaTeX["\text{Add.}"]}}, "l") → text14
  20. LaTeX((("\text{Use the Product Property of Logarithms to show that }{" * productproperty) * "}\text{.} \cr \text{Verify this property mathematically.}")) → text1

AI Drawing Prompt Reference

Create a diagram illustrating the product property of logarithms, log_a(xy) = log_a(x) + log_a(y). Start by defining logarithms, then show the transition from logarithmic form to exponential form step by step. Use specific values like a=10, x=3, y=5 to verify the property, and finally prove it algebraically.
The prompt above was compiled from the original drawing commands and is for reference only.