CCGPS AA 4.2.3 示例 2：利用对数乘法性质证明 log_a(xy) = log_a x + log_a y

GeoGebra 命令及参数

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