Problem: x x
2 x + 2 x + 4 ( x 2 ) + 8 ( x 3 ) + 16 ( x 4 ) = 40 ? log 2 β β x β + log 2 β x + log 4 β ( x 2 ) + log 8 β ( x 3 ) + log 1 6 β ( x 4 ) = 4 0 ?
Answer Choices:
A. 8 8 16 1 6 32 3 2 256 2 5 6 1024 1 0 2 4 Solution:
Rewriting each logarithm in base 2 gives
1 2 2 x 1 2 + 2 x + 2 2 x 2 + 3 2 x 3 + 4 2 x 4 = 40 2 1 β 2 1 β log 2 β x β + log 2 β x + 2 2 log 2 β x β + 3 3 log 2 β x β + 4 4 log 2 β x β = 4 0
Therefore 5 2 x = 40 5 log 2 β x = 4 0 2 x = 8 log 2 β x = 8 x = n 256 x = n 2 5 6 β
OR
For a β 0 a ξ = 0 2 a ( x a ) = y log 2 a β ( x a ) = y 2 a y = x a 2 a y = x a 2 y = x 2 y = x y = 2 x y = log 2 β x 5 2 x = 40 5 log 2 β x = 4 0 2 x = 8 log 2 β x = 8 x = 256 x = 2 5 6 β
The problems on this page are the property of the MAA's American Mathematics Competitions