Is it possible to use logarithms to convert numbers to binary?

Question

I'm a CS freshman and I find the division way of finding a binary number to be a pain. Is it possible to use log to quickly find 24, for instance, in binary?

Answer 1

If you want to use logarithms, you can.

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Define log₂(b) as log(b) / log(2) or ln(b) / ln(2) (they are the same).

Repeat the following:

• Define n as the integer part of log₂(b). There is a 1 in the n^th position in the binary representation of b.

• Set b = b - 2ⁿ

• Repeat first step until b = 0.

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Worked example: Converting 2835 to binary

• log₂(2835) = 11.47.. => n = 11

  The binary representation has a 1 in the 2¹¹ position.

• 2835 - (2¹¹ = 2048) = 787

  log₂(787) = 9.62... => n = 9

  The binary representation has a 1 in the 2⁹ position.

• 787 - (2⁹ = 512) = 275

  log₂(275) = 8.10... => n = 8

  The binary representation has a 1 in the 2⁸ position.

• 275 - (2⁸ = 256) = 19

  log₂(19) = 4.25... => n = 4

  The binary representation has a 1 in the 2⁴ position.

• 19 - (2⁴ = 16) = 3

  log₂(3) = 1.58.. => n = 1

  The binary representation has a 1 in the 2¹ position.

• 3 - (2¹ = 2) = 1

  log₂(1) = 0 => n = 0

  The binary representation has a 1 in the 2⁰ position.

We know the binary representation has 1s in the 2¹¹, 2⁹, 2⁸, 2⁴, 2¹, and 2⁰ positions:

2^     11 10 9 8 7 6 5 4 3 2 1 0
binary  1  0 1 1 0 0 0 1 0 0 1 1

so the binary representation of 2835 is 101100010011.