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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?

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    Pardon for the question. I didn't find it. Thanks – 2017-01-19
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    Its ok. You can edit your question with some new doubts. Like how to calculate log values used in solving this. – 2017-01-19
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    And give reference to that question too. – 2017-01-19
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    http://math.stackexchange.com/questions/1136486/finding-the-first-digit-of-20152015 – 2017-01-19

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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 $7$.

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    Thank you for the answer. But what if you have't got a calculator or a logarithmic table? – 2017-01-19
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    In principle you can approximate the log by hand, but it would be pretty ugly. So go get a calculator! – 2017-01-19
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    Yeah that's really better. Thank you very much. – 2017-01-19