Change of Base of Logarithm/Base 10 to Base e/Form 2
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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:
- $\ln x = \dfrac {\log_{10} x} {\log_{10} e}$
- $\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03 \ldots$
can then be calculated or looked up.
$\blacksquare$