Show-that-log-10-x-1-give-the-number-of-digits-for-x-Z-

Question Number 10040 by FilupSmith last updated on 21/Jan/17

Show that

⌊ \log_{10} (x) + 1 ⌋ give the number of

digits for x \in Z

Answered by mrW1 last updated on 21/Jan/17

l e t x b e a n i n t e g e r w i t h n d i g i t s, t h e n

10^{n - 1} ⩽ x < 10^{n}

\log_{10} 10^{n - 1} ⩽ \log_{10} (x) < \log_{10} 10^{n}

n - 1 ⩽ \log_{10} (x) < n

n ⩽ \log_{10} (x) + 1 < n + 1

\Rightarrow ⌊ \log_{10} (x) + 1 ⌋ = n

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