Largest Integer Expressible by 3 Digits/Logarithm Base 10

Theorem

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

$\ds \map {\log_{10} } {9^{9^9} }$

$=$

$\ds 9^9 \times \log_{10} 9$

$\ds $

$\approx$

$\ds 387 \, 420 \, 489 \times 0\cdotp 95424 \, 25094 \, 393249$

by calculator

$\ds $

$\approx$

$\ds 369 \, 693 \, 099 \cdotp 63157 \, 03685 \, 87876 \, 1$

by calculator

$\blacksquare$

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.