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

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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:

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


Proof

From Change of Base of Logarithm:

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

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

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

as by definition of $\ln x$:

$\ln x := \log_e x$


The Natural Logarithm of 10:

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

can then be calculated or looked up.

$\blacksquare$


Sources