Difference between revisions of "1967 AHSME Problems/Problem 26"

Revision as of 19:22, 10 March 2017

Problem

If one uses only the tabular information $10^3=1000$ , $10^4=10,000$ , $2^{10}=1024$ , $2^{11}=2048$ , $2^{12}=4096$ , $2^{13}=8192$ , then the strongest statement one can make for $\log_{10}{2}$ is that it lies between:

$\textbf{(A)}\ \frac{3}{10} \; \text{and} \; \frac{4}{11}\qquad \textbf{(B)}\ \frac{3}{10} \; \text{and} \; \frac{4}{12}\qquad \textbf{(C)}\ \frac{3}{10} \; \text{and} \; \frac{4}{13}\qquad \textbf{(D)}\ \frac{3}{10} \; \text{and} \; \frac{40}{132}\qquad \textbf{(E)}\ \frac{3}{11} \; \text{and} \; \frac{40}{132}$

Solution

Since $1024$ is greater than $1000$ .

$log 1024 > 3$

$10 * log 2 > 3$

and $log 2 > 3/10$ .

Similarly, $8192 < 10000$ , so $log 8192 < 4$

$13 * log 2 < 4$

and $log 2 < 4/13$

Therefore $3/10 < log 2 < 4/13$ so the answer is $\fbox{C}$

1967 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 25	Followed by Problem 27
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
All AHSME Problems and Solutions

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

@@ Line 9: / Line 9: @@
 == Solution ==
-Since 1024 is greater than 1000.
+Since <math>1024</math> is greater than <math>1000</math>.
-log 1024 > 3
+<math>log 1024 > 3</math>
-* log 2 > 3
+<math>10 * log 2 > 3</math>
-and log 2 > 3/10.
+and <math>log 2 > 3/10</math>.
-Similarly, 8192 < 10000, so log 8192 < 4
+Similarly, <math>8192 < 10000</math>, so <math>log 8192 < 4</math>
-* log 2 < 4
+<math>13 * log 2 < 4</math>
-and log 2 < 4/13
+and <math>log 2 < 4/13</math>
-Therefore  3/10 < log 2 < 4/13
+Therefore  <math>3/10 < log 2 < 4/13</math>
 so the answer is <math>\fbox{C}</math>

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Difference between revisions of "1967 AHSME Problems/Problem 26"

Revision as of 19:22, 10 March 2017

Problem

Solution

See also