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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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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 27, 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 am stands for a × a × . . . × a, where there are m as in the product. This product is called “a to the

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