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Let's say I want to use an alphabet of 6.000 symbols to display 512 Bit of information.

With my rudimentary math skills, I figured out this equation should give the answer:
6000^x = 2^512
(http://www.wolframalpha.com/input/?i=6000%5Ex+%3D+2%5E512)

Thanks to WolframAlpha I got a result: 41 symbols (after a test with 6000^41 > 2^512 ; 6000^40 < 2^512)

But I would like to know...

  1. what this graphs tells me:
    enter image description here
  2. how to change the formula the get the integer '42'.
hdev
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1 Answers1

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Probably $2^{512}$ is too large to be representable by the software to any reliable precision.

The equation$$6000^x = 2^{512}$$

Logarithm of both sides:

$$x\cdot \log(6000) = 512 \cdot \log(2)$$ $$x = \frac{512 \cdot \log(2)}{\log(6000)}= 40.794\cdots\geq 40$$

You are sure to be able to store 40 samples from an alphabet of 6000 symbols.

For second part of the question we switch places of $x$:

$$6000^{42}=2^{x}$$

$$x = \frac{42\log(6000)}{\log(2)} = 527.13\cdots \approx 528$$

So we need 528 bits for 42 symbols.

mathreadler
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  • Since OP wants to express 512 bits of data, he'll need at least $40.794\ldots$, so at least $41$ symbols, no? – Travis Willse Apr 19 '17 at 17:45
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    Actually, the precise value of $2^{512}$, which is 1340780792994259709957402499820584612747936582059239337772356144372176403007354697680187429816690342769003185818648605085375388281194656994643364900608409 is explicitly mentioned in the legend – Hagen von Eitzen Apr 19 '17 at 17:46
  • @Travis I interpreted the question as OP had 512 bits available storage and wondered how many samples that would give him. In that case we should round down because it does not help if we can store $79%$ of that last 41st symbol. – mathreadler Apr 19 '17 at 17:47
  • You would the wolfram alpha syntax look like, to get the result as a decimal? (40.794) – hdev Apr 22 '17 at 17:02
  • @dh_cgn this seems to work: http://www.wolframalpha.com/input/?i=solve+6000%5Ex+-+2%5E%7B512%7D+%3D+0+wrt+x+real you may need to click approximate form – mathreadler Apr 22 '17 at 17:06