Difference between revisions of "1950 AHSME Problems/Problem 44"
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==Problem== | ==Problem== | ||
| − | The graph of <math> y | + | The graph of <math> y=\log x</math> |
| − | <math>\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\ | + | <math>\textbf{(A)}\ \text{Cuts the }y\text{-axis} \qquad\\ |
\textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ | \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ | ||
\textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ | \textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ | ||
\textbf{(D)}\ \text{Cuts neither axis} \qquad\\ | \textbf{(D)}\ \text{Cuts neither axis} \qquad\\ | ||
\textbf{(E)}\ \text{Cuts all circles whose center is at the origin}</math> | \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}</math> | ||
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== Solution == | == Solution == | ||
| − | The domain of <math>\log x</math> is the set of all positive reals, so the graph of <math>y=\log x</math> clearly doesn't cut the <math>y</math>-axis. It therefore doesn't cut every line perpendicular to the <math>x</math>-axis. It does however cut the <math>x</math>-axis at <math>(1,0)</math>. In addition, if one examines the graph of <math>y=\log x</math>, one can clearly see that there are many circles centered at the origin that do not intersect the graph of <math>y=\log x</math>. Therefore the answer is <math>\boxed{\textbf{(C)}\ \text{Cuts the }x\text{-axis}}</math>. | + | The domain of <math>\log x</math> is the set of all <math>\underline{positive}</math> reals, so the graph of <math>y=\log x</math> clearly doesn't cut the <math>y</math>-axis. It therefore doesn't cut every line perpendicular to the <math>x</math>-axis. It does however cut the <math>x</math>-axis at <math>(1,0)</math>. In addition, if one examines the graph of <math>y=\log x</math>, one can clearly see that there are many circles centered at the origin that do not intersect the graph of <math>y=\log x</math>. Therefore the answer is <math>\boxed{\textbf{(C)}\ \text{Cuts the }x\text{-axis}}</math>. |
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== See Also == | == See Also == | ||
Latest revision as of 15:16, 9 May 2015
Problem
The graph of 
Solution
The domain of 
is the set of all 
reals, so the graph of clearly doesn't cut the 
-axis. It therefore doesn't cut every line perpendicular to the 
-axis. It does however cut the -axis at 
. In addition, if one examines the graph of , one can clearly see that there are many circles centered at the origin that do not intersect the graph of . Therefore the answer is 
.
See Also
| 1950 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 43 |
Followed by Problem 45 | |
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| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
