Difference between revisions of "1965 AHSME Problems/Problem 31"
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== 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 08:43, 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 | |
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All AHSME Problems and Solutions |
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