TAOCP 1.2.2 Exercise 12
Equation (8) shows that 10^{0.
Section 1.2.2: Numbers, Powers, and Logarithms
Exercise 12. [02] Explain why Eq. (10) follows from Eqs. (8).
Verified: yes
Solve time: 36s
Equation (8) shows that
$$ 10^{0.30102999}=1.99999999739\ldots $$
and
$$ 10^{0.30103000}=2.0000000199\ldots . $$
Since
$$ 1.99999999739\ldots < 2 < 2.0000000199\ldots, $$
the number $2$ lies strictly between the values of $10^x$ corresponding to the decimal approximations $x=0.30102999$ and $x=0.30103000$.
By the definition of logarithm, Eq. (9), the exponent $x$ for which $10^x=2$ is $\log_{10}2$. Therefore $\log_{10}2$ lies between $0.30102999$ and $0.30103000$, so its decimal expansion begins
$$ \log_{10}2=0.30102999\ldots, $$
which is Eq. (10). ∎