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\noindent {\bf Problem statement} The following statements are facts:

$${\left(10,\!000,\!000,\!000\right)^{\left( 1 \over 10,000,000,000
\right)} \approx 1.\ 00000\ 000\underline{23}\ \underline{02585}\
\underline{09}564}$$

\noindent and $$\ln 10\approx \underline{2}.\underline{30258}\
\underline{509}29$$

\noindent Explain the amazing coincidence of the digits.

\medskip

\noindent {\bf Hint} Approximate $e^x$ when $x$ is small.




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