Change of Base of Logarithm/Base 2 to Base 10

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:

$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right)$

The common logarithm of $2$:

$\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

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: $(14)$