Difference between revisions of "1970 AHSME Problems/Problem 8"
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== Problem == | == Problem == | ||
− | If <math>a=log_8 | + | If <math>a=\log_8 225</math> and <math>b=\log_2 15</math>, then |
<math>\text{(A) } a=b/2\quad | <math>\text{(A) } a=b/2\quad | ||
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== Solution == | == Solution == | ||
− | <math>\fbox{B}</math> | + | The solutions imply that finding the ratio <math>\frac{a}{b}</math> will solve the problem. We compute <math>\frac{a}{b}</math>, use change-of-base to a neutral base, rearrange the terms, and then use the reverse-change-of-base: |
+ | |||
+ | <math>\frac{\log_8 225}{\log_2 15}</math> | ||
+ | |||
+ | <math>\frac{\frac{\ln 225}{\ln 8}}{\frac{\ln 15}{\ln 2}}</math> | ||
+ | |||
+ | <math>\frac{\ln 225 \ln 2}{\ln 15 \ln 8}</math> | ||
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+ | <math>\frac{\ln 225}{\ln 15} \cdot \frac{\ln 2}{\ln 8}</math> | ||
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+ | <math>\ln_{15} 225 \cdot \ln_8 2</math> | ||
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+ | Since <math>15^2 = 225</math>, the first logarithm is <math>2</math>. Since <math>8^{\frac{1}{3} = 2</math>, the second logarithm is <math>\frac{1}{3}</math>. | ||
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+ | Thus, we have <math>\frac{a}{b} = \frac{2}{3}</math>, or <math>a = \frac{2}{3}b</math>, which is option <math>\fbox{B}</math>. | ||
== See also == | == See also == |
Revision as of 20:02, 13 July 2019
Problem
If and , then
Solution
The solutions imply that finding the ratio will solve the problem. We compute , use change-of-base to a neutral base, rearrange the terms, and then use the reverse-change-of-base:
Since , the first logarithm is . Since $8^{\frac{1}{3} = 2$ (Error compiling LaTeX. ! Missing } inserted.), the second logarithm is .
Thus, we have , or , which is option .
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.