Difference between revisions of "1983 AHSME Problems/Problem 25"
(→Solution) |
(→Solution 2) |
||
Line 16: | Line 16: | ||
== Solution 2 == | == Solution 2 == | ||
− | We have <math>60^a = 3</math> and <math>60^b = 5</math>. We can say that <math>a = log_{60} 3</math> and <math>b = log_{60} 5</math>. | + | We have <math>60^a = 3</math> and <math>60^b = 5</math>. We can say that <math>a = \log_{60} 3</math> and <math>b = \log_{60} 5</math>. |
<cmath>12^{(1-a-b)/[2(1-b)]} = 12^{(1-(a+b))/[2(1-b)]}</cmath> | <cmath>12^{(1-a-b)/[2(1-b)]} = 12^{(1-(a+b))/[2(1-b)]}</cmath> | ||
− | We can evaluate (a+b) by the Addition Identity for Logarithms, <math>(a+b) = log_{60} 15</math>. Also, <math>1 = log_{60} 60</math>. | + | We can evaluate (a+b) by the Addition Identity for Logarithms, <math>(a+b) = \log_{60} 15</math>. Also, <math>1 = \log_{60} 60</math>. |
− | <cmath> (1-(a+b) = log_{60} 60 - log_{60} 15 = log_{60} 4 </cmath> | + | <cmath> (1-(a+b) = \log_{60} 60 - \log_{60} 15 = \log_{60} 4 </cmath> |
− | Now we have to evaluate [2(1-b)], the denominator of the fractional exponent. Once again, we can say <math>1 = log_{60} 60</math> | + | Now we have to evaluate [2(1-b)], the denominator of the fractional exponent. Once again, we can say <math>1 = \log_{60} 60</math> |
− | <cmath> 2(1-b) = 2(log_{60} 12)</cmath> | + | <cmath> 2(1-b) = 2(\log_{60} 12)</cmath> |
− | <cmath>12^{(log_{60} 4)/[2(log_{60} 12]} = 12^{\frac{1}{2} \cdot log_{12} 4} = 4^{1/2} = 2</cmath> | + | <cmath>12^{(\log_{60} 4)/[2(\log_{60} 12]} = 12^{\frac{1}{2} \cdot \log_{12} 4} = 4^{1/2} = 2</cmath> |
+ | |||
+ | ~YBSuburbanTea | ||
==See Also== | ==See Also== |
Latest revision as of 11:12, 14 January 2022
Contents
Problem 25
If and , then is
Solution
We have that . We can substitute our value for 5, to get Hence Since , we have Therefore, we have
Solution 2
We have and . We can say that and .
We can evaluate (a+b) by the Addition Identity for Logarithms, . Also, .
Now we have to evaluate [2(1-b)], the denominator of the fractional exponent. Once again, we can say
~YBSuburbanTea
See Also
1983 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Problem 26 | |
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.