In a certain logarithmic system, $\log 450$ is 1.1639 greater than $\log 40$. What is the base of the system and what are the logarithms of the two numbers in this system?
Let's denote the base of the sought system by $x$. According to the problem,

$$
\stackrel{x}{\log } 450-\stackrel{x}{\log } 40=1.1639
$$

which means

$$
\stackrel{x}{\log } \frac{450}{40}=1.1639
$$

or

$$
x^{1.1639}=\frac{450}{40}
$$

Taking the logarithm of both sides with base 10,

$$
1.1639 \lg x=\lg 450-\lg 40
$$

from which

$$
\lg x=\frac{\lg 450-\lg 40}{1.1639}=0.9031
$$

and thus

$$
x=8
$$

If

$$
\stackrel{8}{\log } 40=y, \text{ then } 8^{y}=40
$$

or

$$
y=\frac{\lg 40}{\lg 8}=1.774
$$

and thus

$$
\stackrel{8}{\log } 450=\stackrel{8}{\log } 40+1.1639=2.9379
$$

Kecskés Kornél (Esztergom, I. István g. III. o. t.)