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Difference between revisions of "1950 AHSME Problems/Problem 18"

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== Problem==
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== Problem ==
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Of the following
  
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Of the following
 
 
(1) <math> a(x-y)=ax-ay </math>
 
(1) <math> a(x-y)=ax-ay </math>
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(2) <math> a^{x-y}=a^x-a^y </math>
 
(2) <math> a^{x-y}=a^x-a^y </math>
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(3) <math> \log (x-y)=\log x-\log y </math>
 
(3) <math> \log (x-y)=\log x-\log y </math>
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(4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math>
 
(4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math>
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(5) <math> a(xy)=ax\times ay </math>
 
  
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<math> \textbf{(A)}\ \text{Only 1 and 4 are true}\qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\ \textbf{(E)}\ \text{Only 1 is true} </math>
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(5) <math> a(xy)=ax \cdot ay </math>
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<math>\textbf{(A)}\text{Only 1 and 4 are true}\qquad\\\textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\\textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\\textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\\textbf{(E)}\ \text{Only 1 is true} </math>
  
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==Solution==
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== Solution ==
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The distributive property doesn't apply to logarithms or in the ways illustrated, and only applies to addition and subtraction. Also, <math>a^{x-y} = \frac{a^x}{a^y}</math>, so <math>\boxed{\textbf{(E)} \text{ Only 1 is true}}</math>.
  
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The distributive property doesn't apply to logarithms or in the ways illustrated, and only applies to addition and subtraction. Also, <math>a^{x-y} = \frac{a^x}{a^y}</math>, so <math>\textbf{(E)} \text{Only 1 is true}</math>
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== See Also ==
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==See Also==
 
  
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{{AHSME box|year=1950|num-b=17|num-a=19}}
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{{AHSME 50p box|year=1950|num-b=17|num-a=19}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 21:19, 6 February 2023

Problem

Of the following

(1) $a(x-y)=ax-ay$

(2) $a^{x-y}=a^x-a^y$

(3) $\log (x-y)=\log x-\log y$

(4) $\frac{\log x}{\log y}=\log{x}-\log{y}$

(5) $a(xy)=ax \cdot ay$

$\textbf{(A)}\text{Only 1 and 4 are true}\qquad\\\textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\\textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\\textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\\textbf{(E)}\ \text{Only 1 is true}$

Solution

The distributive property doesn't apply to logarithms or in the ways illustrated, and only applies to addition and subtraction. Also, $a^{x-y} = \frac{a^x}{a^y}$, so $\boxed{\textbf{(E)} \text{ Only 1 is true}}$.

See Also

1950 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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