Is there a way to prove log 2 base 10 <= 0.301 other than verifying the value using a calculator? Please give a detailed explanation, if proof is possible.
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Not an answer, but we can cheaply get a good approximation. Since $2^{10}$ is a little bigger than $10^3$, but not much bigger, we know that $10\log_{10} 2$ is a little bigger than $3$, so $\log_{10} 2$ is a little bigger than $0.3$. – André Nicolas Apr 13 '16 at 06:32
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Yes an approximate let $\frac{1+x}{1-x}=2$ thus $x=1/3$ and then using taylor series for $log(\frac{1+x}{1-x})=2(x+\frac{x^3}{3!}+..)$
Archis Welankar
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