# Change of Base of Logarithm/Base e to Base 10/Form 1

## Theorem

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

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

Then:

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

## Proof

$\log_a x = \log_a b \, \log_b x$

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

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

can be calculated or looked up.

$\blacksquare$