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

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Of the following  
 
Of the following  
 +
 
(1) <math> a(x-y)=ax-ay </math>
 
(1) <math> a(x-y)=ax-ay </math>
 +
 
(2) <math> a^{x-y}=a^x-a^y </math>
 
(2) <math> a^{x-y}=a^x-a^y </math>
 +
 
(3) <math> \log (x-y)=\log x-\log y </math>
 
(3) <math> \log (x-y)=\log x-\log y </math>
 +
 
(4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math>
 
(4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math>
 +
 
(5) <math> a(xy)=ax\times ay </math>
 
(5) <math> a(xy)=ax\times ay </math>
 +
 +
  
 
<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>
 
<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>

Revision as of 20:27, 27 January 2015

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) $a(xy)=ax\times ay$


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

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 $\textbf{(E)} \text{ Only 1 is true}$

See Also

1950 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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All AHSME Problems and Solutions

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