The equation [tex]2 = 1[/tex] shows the result. (Correct choice: A)
How to analyze a power function
In this question we must apply algebra properties for power functions and definition of logarithms to simplify given expression and make conclusions from results:
1) [tex]\left(\frac{1}{5000} \right)^{2\cdot z}\cdot 5000^{2\cdot z + 2} = 5000[/tex] Given
2) [tex](5000^{-1})^{2\cdot z}\cdot 5000^{2\cdot z + 2} = 5000[/tex] Definition of division
3) [tex]5000^{-2\cdot z}\cdot 5000^{2\cdot z + 2} = 5000[/tex] [tex](x^{m})^{n} = x^{m\cdot n}[/tex]/[tex](-a)\cdot b = -a\cdot b[/tex]
4) [tex]5000^{2} = 5000[/tex] [tex]x^m \cdot x^{n} = x^{m+n}[/tex]
5) [tex]\log_{5000} 5000^{2} = \log_{5000} 5000[/tex] Definition of logarithm
6) [tex]2\cdot \log_{5000} 5000 = \log_{5000} 5000[/tex] [tex]\log_{b} a^{n} = n\cdot \log_{b} a[/tex]
7) [tex]2 = 1[/tex] Compatibility with multiplication/Existence of multiplicative inverse/Modulative property/Result
The equation [tex]2 = 1[/tex] shows the result. (Correct choice: A) [tex]\blacksquare[/tex]
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