Difference between revisions of "1965 AHSME Problems/Problem 31"
(created solution page) |
(→Solution) |
||
| Line 10: | Line 10: | ||
== Solution == | == Solution == | ||
| − | <math>\fbox{C}</math> | + | |
| + | From the properties of [[logarithms]], we know that <math>(\log_ax)(\log_xb)=\log_ab</math>, and <math>\log_xb=\frac{1}{\log_bx}</math>. Thus, to satisfy the given equation, we are looking for values of <math>x</math> when <math>\log_bx=\frac{1}{\log_bx}</math>, or when <math>\log_bx=\pm 1</math>. This happens either when <math>x=b</math> or <math>x=\frac{1}{b}</math>. Thus we have <math>2</math> solutions, consistent with answer choice <math>\fbox{\textbf{(C)}}</math>. | ||
== See Also == | == See Also == | ||
Latest revision as of 09:43, 19 July 2024
Problem
The number of real values of 
satisfying the equality 
, where 
, is:
Solution
From the properties of logarithms, we know that 
, and 
. Thus, to satisfy the given equation, we are looking for values of when 
, or when 
. This happens either when 
or 
. Thus we have 
solutions, consistent with answer choice 
.
See Also
| 1965 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 30 |
Followed by Problem 32 | |
| 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 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
