Difference between revisions of "2002 AMC 12P Problems/Problem 21"
(→Problem) |
(→Problem) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
Let <math>a</math> and <math>b</math> be real numbers greater than <math>1</math> for which there exists a positive real number <math>c,</math> different from <math>1</math>, such that | Let <math>a</math> and <math>b</math> be real numbers greater than <math>1</math> for which there exists a positive real number <math>c,</math> different from <math>1</math>, such that | ||
| + | |||
<cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath> | <cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath> | ||
| + | |||
Find the largest possible value of <math>\log_a b.</math> | Find the largest possible value of <math>\log_a b.</math> | ||
Revision as of 00:09, 30 December 2023
Problem
Let 
and 
be real numbers greater than 
for which there exists a positive real number 
different from , such that
![\[2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.\]](http://latex.artofproblemsolving.com/d/3/c/d3cf084efbe6ea20e4571491e473bcc3aa15c877.png)
Find the largest possible value of 
Solution
If 
, then 
. Since 
, must be 
to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of 
.
See also
| 2002 AMC 12P (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
