TAOCP 1.2.2 Exercise 20
Yes.
Section 1.2.2: Numbers, Powers, and Logarithms
Exercise 20. [10] Is there any simple relation between $\log_{10} 2$ and $\log_2 10$?
Verified: yes
Solve time: 36s
Yes. By the change-of-base formula (14),
$$ \log_2 10=\frac{\log_{10}10}{\log_{10}2}. $$
Since $\log_{10}10=1$, this becomes
$$ \log_2 10=\frac{1}{\log_{10}2}. $$
Therefore the two quantities are reciprocals:
$$ (\log_{10}2)(\log_2 10)=1. $$
Using Eq. (10),
$$ \log_{10}2=0.30102999\ldots, $$
hence
$$ \log_2 10=\frac{1}{0.30102999\ldots}=3.32192809\ldots. $$
Thus
$$ \boxed{\log_2 10=\frac{1}{\log_{10}2}}. $$