1968 AHSME Problems/Problem 29
Contents
[hide]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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