Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1983 AHSME Problems/Problem 25"

m (Improved clarity)
(→‎Solution 2)
 
(5 intermediate revisions by 3 users not shown)
Line 12: Line 12:
 
<cmath>4=\frac{60}{60^a 60^b}=60^{1-a-b}.</cmath>
 
<cmath>4=\frac{60}{60^a 60^b}=60^{1-a-b}.</cmath>
 
Therefore, we have
 
Therefore, we have
−
<cmath>60^{(1-a-b)/2}=4^{1/2}=\boxed{\textbf{(B)} 2}</cmath>
+
<cmath>60^{(1-a-b)/2}=4^{1/2}=\boxed{\textbf{(B)}\ 2}</cmath>
 +
 
 +
 
 +
== 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>.
 +
 
 +
<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>.
 +
 
 +
<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>
 +
 
 +
<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>
 +
 
 +
~YBSuburbanTea
  
 
==See Also==
 
==See Also==

Latest revision as of 11:12, 14 January 2022

Problem 25

If $60^a=3$ and $60^b=5$, then $12^{(1-a-b)/\left(2\left(1-b\right)\right)}$ is

$\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 2\sqrt{3}\qquad$

Solution

We have that $12=\frac{60}{5}$. We can substitute our value for 5, to get \[12=\frac{60}{60^b}=60\cdot 60^{-b}=60^{1-b}.\] Hence \[12^{(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-b)(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-a-b)/2}.\] Since $4=\frac{60}{3\cdot 5}$, we have \[4=\frac{60}{60^a 60^b}=60^{1-a-b}.\] Therefore, we have \[60^{(1-a-b)/2}=4^{1/2}=\boxed{\textbf{(B)}\ 2}\]


Solution 2

We have $60^a = 3$ and $60^b = 5$. We can say that $a = \log_{60} 3$ and $b = \log_{60} 5$.

\[12^{(1-a-b)/[2(1-b)]} = 12^{(1-(a+b))/[2(1-b)]}\]

We can evaluate (a+b) by the Addition Identity for Logarithms, $(a+b) = \log_{60} 15$. Also, $1 = \log_{60} 60$.

\[(1-(a+b) = \log_{60} 60 - \log_{60} 15 = \log_{60} 4\]

Now we have to evaluate [2(1-b)], the denominator of the fractional exponent. Once again, we can say $1 = \log_{60} 60$

\[2(1-b) = 2(\log_{60} 12)\]

\[12^{(\log_{60} 4)/[2(\log_{60} 12]} = 12^{\frac{1}{2} \cdot \log_{12} 4} = 4^{1/2} = 2\]

~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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1983_AHSME_Problems/Problem_25&oldid=169881"