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 $\ln x$:
- $\ln x := \log_e x$
- $\ln 10 = 2 \cdotp 30258 \, 50929 \, 94 \ldots$
can then be calculated or looked up.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $16$