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What is the best/most correct way to represent the logarithm of a number? Example: $$-3 \log⁡2+5 \log⁡175+2 \log⁡7429+3 \log⁡34749$$

  1. Just leave it the way it was calculated $$-3 \log⁡2+5 \log⁡175+2 \log⁡7429+3 \log⁡34749$$
  2. As a single $\log$ $$\log \biggl(\frac{380082516906650443140753544921875}{8}\biggl)$$
  3. Two logarithms for the positive and the negative part (in case they both exist. Otherwise use the above) $$\log 380082516906650443140753544921875 - \log 8$$
  4. As a sum of logarithms of prime numbers $$-3 \log2 + 15 \log3 + 10 \log5 + 5 \log7 + 3 \log11 + 3 \log13 + 2 \log17 + 2 \log19 + 2 \log23$$
  5. As a sum of logarithms with different coefficients ($a\log b$ means $b$ is the product of the primes numbers with exponent $a$ in the prime factorization) $$-3 \log2 + 15 \log3 + 10 \log5 + 5 \log7 + 3 \log143 + 2 \log7429$$

Is any of these the best way? Does it matter?

EDIT: I got curious because this number is really big. Obviously if the number is smaller the 2nd or 3rd options are fine

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    $\begingroup$ I personally prefer $$\log \biggl(\frac{380082516906650443140753544921875}{8}\biggl)$$ but a nice feature of mathematics is that you can use any of the alternatives you mentioned and they all mean the same thing. $\endgroup$ – Maximilian Janisch Aug 15 at 10:05
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Different representations, while referring to the same number, convey information differently. The context in which this number arises will be your best guide.

Sometimes, you want a sense of how large this number is (such as if it appears as some measurement), in which case something like $72.94$ (I used WolframAlpha to compute this, assuming $\log$ is the natural log; if you want the base-10 log it's more like $31.68$) would actually be most appropriate. Sometimes, you just want that the number exists, in which case you can leave it at 1.

Choice 2 is best if you intend to exponentiate the answer later on. If you want number-theoretic information, such as if you wish to exponentiate this later on or combine it with logs, then I am partial to 4. Choices 3 and 5 strike me as more style choices, but at the expense of information.

At the end of the day, of course, it's up to you--my remarks above are just a guide, and steeped in personal experiences that may not apply to you!

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    $\begingroup$ Nice first answer, Paco! $\endgroup$ – John Hughes Aug 15 at 11:51
  • $\begingroup$ Thank you so much, John! $\endgroup$ – Paco Adajar Aug 15 at 13:21

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