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)=ax-ay </math>
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(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>
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==Solution==
 
==Solution==
  
The distributive property doesn't apply to logarithms in the ways illustrated, and only applies to addition and subtraction. Therefore <math>\boxed{\mathrm{(D)}\text{ only }1\text{ and }2\text{ are 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>
 
 
 
==See Also==
 
==See Also==
  
 
{{AHSME box|year=1950|num-b=17|num-a=19}}
 
{{AHSME box|year=1950|num-b=17|num-a=19}}

Revision as of 10:35, 29 December 2011

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