Difference between revisions of "1953 AHSME Problems/Problem 39"
(/Solution/) |
(→Solution) |
||
(3 intermediate revisions by one other user not shown) | |||
Line 1: | Line 1: | ||
+ | ==Problem== | ||
+ | |||
+ | The product, <math>\log_a b \cdot \log_b a</math> is equal to: | ||
+ | |||
+ | <math>\textbf{(A)}\ 1 \qquad | ||
+ | \textbf{(B)}\ a \qquad | ||
+ | \textbf{(C)}\ b \qquad | ||
+ | \textbf{(D)}\ ab \qquad | ||
+ | \textbf{(E)}\ \text{none of these} </math> | ||
+ | |||
+ | ==Solution== | ||
<cmath>a^x=b</cmath> | <cmath>a^x=b</cmath> | ||
<cmath>b^y=a</cmath> | <cmath>b^y=a</cmath> | ||
Line 4: | Line 15: | ||
<cmath>xy=1</cmath> | <cmath>xy=1</cmath> | ||
<cmath>\log_a b\log_b a=1</cmath> | <cmath>\log_a b\log_b a=1</cmath> | ||
− | As a result, the answer should be 1 \boxed{A}. | + | As a result, the answer should be <math>\boxed{\textbf{(A) }1}</math>. |
+ | |||
+ | ==Solution 2== | ||
+ | Apply the change of base formula to <math>\log_a b</math> and <math>\log_b a</math>. For simplicity, let us convert to base-10 log. | ||
+ | By change of base, the expression becomes <math>\frac{\log b}{\log a} * \frac{\log a}{\log b} = \boxed{\textbf{(A) }1}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME 50p box|year=1953|num-b=38|num-a=40}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 16:57, 22 April 2020
Contents
[hide]Problem
The product, is equal to:
Solution
As a result, the answer should be .
Solution 2
Apply the change of base formula to and . For simplicity, let us convert to base-10 log. By change of base, the expression becomes .
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 38 |
Followed by Problem 40 | |
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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.