\section*{Exercise 3 - 171033}

Jens, Uwe, Dirk, and Peter are discussing which number set the number \(z\), defined by the term

\[
z=\frac{\lg (7-4 \sqrt{3})}{\lg (2-\sqrt{3})}
\]

belongs to.

Jens says that \(z\) is a natural number; Dirk thinks that the number \(z\) is a rational number; Uwe believes that \(z\) is irrational, and Peter suspects that the term does not define any number \(z\) at all.

Determine who is right!
}

It holds that \(2-\sqrt{3}>0\), so \(\lg (2-\sqrt{3})\) is defined. Furthermore,

\[
(2-\sqrt{3})^{2}=4-4 \sqrt{3}+3=7-4 \sqrt{3}
\]

thus \(7-4 \sqrt{3}>0\) and consequently \(\lg (7-4 \sqrt{3})\) is defined. Moreover, \(2-\sqrt{3} \neq 1\), so \(\lg (2-\sqrt{3}) \neq 0\); thus, the given term defines a number \(z\), and for it, it follows that

\[
z=\frac{\lg (2-\sqrt{3})^{2}}{\lg (2-\sqrt{3})} \quad \text { hence } \quad z=\frac{2 \lg (2-\sqrt{3})}{\lg (2-\sqrt{3})}=2
\]

Therefore, Jens and Dirk are correct, while Uwe and Peter are not.