### III.25 The Exponential and Logarithmic Functions

### 1 Exponentiation

The following is a very well-known mathematical sequence: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, . . . . Each term in this sequence is twice the term before, so, for instance, 128, the seventh term in the sequence, is equal to 2 × 2 × 2 × 2 × 2 × 2 × 2. Since repeated multiplications of this kind occur throughout mathematics, it is useful to have a less cumbersome notation for them, so 2 × 2 × 2 × 2 × 2 × 2 × 2 is normally written as 2^{7}, which we read as “2 to the power 7” or just “2 to the 7.” More generally, if *a* is any real number and *m* is any positive integer, then *a*^{m} stands for *a × a × . . . × a,* where there are *m* as in the product. This product is called *“a* to the