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x = log2(y) is defined to be the solution for x to equation y = 2x. For example, since 23 = 8, log2(8) = 3. Here are a few logarithms.
| x | log2(x) |
|---|---|
| 16 | 4 |
| 32 | 5 |
| 64 | 6 |
| 128 | 7 |
| 256 | 8 |
| 512 | 9 |
| 1024 | 10 |
| 1,000,000 | ~20 |
| 1,000,000,000 | ~30 |
One way to compute an approximate logarithm of an integer x is to start with x and then perform steps where, at each step, you take half of the result of the previous step. But if you get a result that is not an integer, then round down. Stop when you reach 1. Then count the number of halving steps that you did. If you did h halving steps, then h is the largest integer that is ≤ log2(x). For example,
1000
500
250
125
62
31
15
7
3
1
involves 9 halving steps, so 9 ≤ log2(1000) ≤ 10.
In fact, log2(1000) is just a little less than 10.
In this course, we write log(n) to mean log2(n).
What is log2(32)? Answer
What is log2(128)? Answer
What is the largest integer n such that n < log2(20)? Answer
What is the largest integer n such that n < log2(150)? Answer
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