Change of Base of Logarithm/Base 2 to Base 10
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Theorem
Let $\log_{10} x$ be the common (base $10$) logarithm of $x$.
Let $\lg x$ be the binary (base $2$) logarithm of $x$.
Then:
- $\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right) = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots \lg x$
Proof
From Change of Base of Logarithm:
- $\log_a x = \log_a b \ \log_b x$
Substituting $a = 10$ and $b = 2$ gives:
- $\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right)$
- $\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can 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: $(14)$