Change of Base of Logarithm/Base 10 to Base e

Theorem

Let $\ln x$ be the natural (base $e$) logarithm of $x$.

Let $\log_{10} x$ be the common (base $10$) logarithm of $x$.

Form 1

$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$

Form 2

$\ln x = \dfrac {\log_{10} x} {\log_{10} e} = \dfrac {\log_{10} x} {0 \cdotp 43429 \, 44819 \, 03 \ldots}$