2014 AMC 12B Problems/Problem 20
Problem
For how many positive integers is ?
Solution
The domain of the LHS implies that Begin from the left hand side Hence, we have integers from 41 to 49 and 51 to 59. There are integers.
Solution 2
This solution is similar to the first solution, but perhaps more clear.
By the properties of logarithms, Therefore, We can see that this function is concave down (that is, it looks like an upside-down U shape); that is, it has a maximum value. This maximum value is attained at , in which This is the one point on the function that does not satisfy the given condition.
We also know that, as noted in solution 1, must be greater than and less than , since otherwise, or would be undefined. Thus, the possible values of are the positive integers from to inclusive, excluding Thus, there are total integers.
~ Technodoggo
Video Solution by OmegaLearn
https://youtu.be/RdIIEhsbZKw?t=1088
~ pi_is_3.14
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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 | |
All AMC 12 Problems and Solutions |
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