Find the first digit (the left one) of the number $2016^{2016}$, not by actually compute it. I know the solution is 7, thanks to Wolfram Alpha's power, but I did not succeeded in finding it.

Question number two: how may i calculate log values used in solving this?

  • $\begingroup$ Pardon for the question. I didn't find it. Thanks $\endgroup$ – Blumer Jan 19 '17 at 18:59
  • $\begingroup$ Its ok. You can edit your question with some new doubts. Like how to calculate log values used in solving this. $\endgroup$ – Kanwaljit Singh Jan 19 '17 at 19:00
  • $\begingroup$ And give reference to that question too. $\endgroup$ – Kanwaljit Singh Jan 19 '17 at 19:01
  • $\begingroup$ math.stackexchange.com/questions/1136486/… $\endgroup$ – Kanwaljit Singh Jan 19 '17 at 19:01

The leftmost digit of a positive integer $x$ is $d$ if the fractional part of $\log_{10}(x)$ is in the interval $[\log_{10}(d), \log_{10}(d+1))$.

Lin this case $\log_{10}(2016^{2016}) = 2016 \log_{10}(2016) \approx 6661.8529$. The fractional part $0.8529\ldots$ is between $\log_{10}(7) \approx .8451$ and $\log_{10}(8) \approx 0.9031$, so the first digit is indeed $7$.

  • $\begingroup$ Thank you for the answer. But what if you have't got a calculator or a logarithmic table? $\endgroup$ – Blumer Jan 19 '17 at 18:57
  • $\begingroup$ In principle you can approximate the log by hand, but it would be pretty ugly. So go get a calculator! $\endgroup$ – Robert Israel Jan 19 '17 at 19:05
  • $\begingroup$ Yeah that's really better. Thank you very much. $\endgroup$ – Blumer Jan 19 '17 at 19:07

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