Find the number of digits of $2013^{2013}$?

Question

Is is possible to find the number of digits of $2013^{2013}$ without a calculator?

Answer 1

HINT:

Number of digits of $N$ in base $b$ =$1+\lfloor\log_bN\rfloor$ [Proof]

Without calculator $10^3<2013<10^4\implies 3<\log_{10}2013<4$

or as $\sqrt{10^7}>3000>2013, 10^7>2013^2\implies \log_{10}2013<\frac72=3.5$

Answer 2

$$1+\lfloor\log_{10}(2013^{2013})\rfloor = 1 + \lfloor 2013\log_{10}(2013)\rfloor = 6651$$

A good estimate without using a calculator is given by $$\log_{10}(2013) = \log_{10}(1000\cdot2\cdot1.0065) = \log_{10}(1000) + \log_{10}(2)+\log_{10}(1.0065)$$ which is approximatively $$\log_{10}(2013)\approx 3+\frac{1}{3}+0 \approx 3.35$$ where the exact result is $\log_{10}(2013) = 3.30384\ldots$