the-number-of-digits-of-a-power-of-2

The number of digits of a power of 2

What is the number of digits of a power of 2?
2 ---> 1 digit
2 * 2 = 4 ---> 1 digit
2 * 2 * 2 = 8 ---> 1 digit
2 * 2 * 2 * 2 = 16 ---> 2 digits
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2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024 ---> 4 digits
Then, given the exponent, what would be the number of digits of that power?