1968 AHSME Problems/Problem 29
Contents
Problem
Given the three numbers 
with 
. Arranged in order of increasing magnitude, they are:
Solution 1
Seeing that we need to compare values with exponents, we think logarithms. Taking the logarithm base 
of each term, we obtain 
, , and 
. Because 
, 
is monotonically decreasing, so the order of terms by magnitude in our new set of numbers will be reversed compared to the original set (i.e. if 
, then 
. However, the order of this set will be reversed again (back to the order of the original set) when we take the logarithm base a second time. After doing this operation, we find the values 
, , and , which correspond to , 
, and 
, respectively. Because 
, , and so, by the correspondence detailed above, 
, which yields us answer choice 
.
Solution 2
Because , taking to the th power will bring it closer to , thereby raising its value. Because we have established that 
, and 
is a monotonically decreasing function, we know that 
. However, because 
, 
, compared to , will be closer to (but still less than) . Thus, 
. Putting this all together, we see that 
, or , which is answer choice .
See also
| 1968 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 28 |
Followed by Problem 30 | |
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