Change of Base of Logarithm/Base 10 to Base e/Form 2

Theorem

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

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

Then:

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

Proof

From Change of Base of Logarithm:

$\log_a x = \dfrac {\log_b x} {\log_b a}$

Substituting $a = e$ and $b = 10$ gives:

$\ln x = \dfrac {\log_{10} x} {\log_{10} e}$

The Common Logarithm of e:

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

can then be calculated or looked up.

$\blacksquare$