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

(Problem)
 
(7 intermediate revisions by 5 users not shown)
Line 1: Line 1:
== Problem==
+
== Problem ==
 +
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>
 
  
<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>
+
(5) <math> a(xy)=ax \cdot 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>
 +
 
 +
== Solution ==
 +
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>.
  
==Solution==
+
== See Also ==
  
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>
+
{{AHSME 50p box|year=1950|num-b=17|num-a=19}}
==See Also==
 
  
{{AHSME box|year=1950|num-b=17|num-a=19}}
+
[[Category:Introductory Algebra Problems]]
 +
{{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 (ProblemsAnswer KeyResources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png