Largest Integer Expressible by 3 Digits/Logarithm Base 10

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Theorem

$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$


Proof

\(\displaystyle \map {\log_{10} } {9^{9^9} }\) \(=\) \(\displaystyle 9^9 \times \log_{10} 9\)
\(\displaystyle \) \(\approx\) \(\displaystyle 387 \, 420 \, 489 \times 0\cdotp 95424 \, 25094 \, 393249\) by calculator
\(\displaystyle \) \(\approx\) \(\displaystyle 369 \, 693 \, 099 \cdotp 63157 \, 03685 \, 87876 \, 1\) by calculator

$\blacksquare$


Historical Note

Horace Scudder Uhler published the value of $\map {\log_{10} } {9^{9^9} }$ to $250$ decimal places in $1947$.

Apparently he found this sort of calculation relaxing.


Sources