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

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:

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

Proof

From Change of Base of Logarithm:

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

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

$\ln x = \paren {\ln 10} \paren {\log_{10} x}$

The Natural Logarithm of 10:

$\ln 10 = 2 \cdotp 30258 \, 50929 \, 94 \ldots$

can be calculated or looked up.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Change of Base of Logarithms: $7.14$