Difference between revisions of "2015 AMC 12A Problems/Problem 14"
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==Problem== | ==Problem== | ||
− | What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>? | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
<math> \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36</math> | <math> \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36</math> | ||
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which becomes after combining: | which becomes after combining: | ||
<cmath>\log_a 24 = 1.</cmath> | <cmath>\log_a 24 = 1.</cmath> | ||
+ | |||
Hence <math>a = 24</math>, and the answer is <math>\textbf{(D)}.</math> | Hence <math>a = 24</math>, and the answer is <math>\textbf{(D)}.</math> | ||
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+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/RdIIEhsbZKw?t=482 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution== | ||
+ | Video starts at 1:04, uses the above solution: https://www.youtube.com/watch?v=OJyWHzjiu2A&t=44s | ||
== See Also == | == See Also == | ||
{{AMC12 box|year=2015|ab=A|num-b=13|num-a=15}} | {{AMC12 box|year=2015|ab=A|num-b=13|num-a=15}} |
Latest revision as of 15:46, 21 August 2024
Problem
What is the value of for which
?
Solution
We use the change of base formula to show that
Thus, our equation becomes
which becomes after combining:
Hence , and the answer is
Video Solution by OmegaLearn
https://youtu.be/RdIIEhsbZKw?t=482
~ pi_is_3.14
Video Solution
Video starts at 1:04, uses the above solution: https://www.youtube.com/watch?v=OJyWHzjiu2A&t=44s
See Also
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 | |
All AMC 12 Problems and Solutions |