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

From Change of Base of Logarithm:

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

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

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

The Common Logarithm of e:

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

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: $7.15$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,43429 44819 \ldots$