Suppose that logk is the arithmetic mean of log1,log2,log4,…,log(2n−1). Then which of these is true:
- k=22n(n−1)
- log2k=2n(n−1)
- logk4=1−n4
- k=(2)n−1
- k=n22n2n−1
Solution
Using the laws of logs and exponents:
logk======n1(log1+log2+log4+…+log(2n−1))n1(0+log21+log22+…+log(2n−1))n1log(21⋅22⋅…⋅2n−1)n1log(21+2+⋯+(n−1))nlog2⋅(1+2+⋯+(n−1))nlog2⋅2n(n−1)=2(n−1)log2
Therefore,
k=102(n−1)log2=(10log2)2n−1=22n−1=(2)n−1.
The answer is D.