I'm reading Rosen's Discrete Mathematics and Its Application, at Page 212, it's about the "Big-O" notation using in computer science.
This is the description in the book:
And here is my reasoning:
Since there wasn't a program with "negative" steps, I only consider the case $n>0$.
With $n$ is $O(2^n)$, there exist two constants $C$ and $k$ such that $$n \le C \cdot 2^n, \text{when } n>k.$$
Now take logarithms on both side we have $$\log{n} \le \log{C} + n$$
Multiply both side with $d$, we get $$d \cdot \log{n} \le d\cdot \log{C} + d \cdot n,$$
Then I stuck.
Please give me some hints about how to do next... Thanks a lot!