Change of Base of Logarithm/Base 10 to Base 2
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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:
- $\lg x = \dfrac {\log_{10} x} {\log_{10} 2} = \dfrac {\log_{10} x} {0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots}$
Proof
From Change of Base of Logarithm:
- $\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = e$ and $b = 10$ gives:
- $\log_e x = \dfrac {\log_{10} x} {\log_{10} e}$
- $\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can then be calculated or looked up.
$\blacksquare$