\section*{Exercise 8 - V01208}

A fallacy "Two is greater than four!"

Obviously, we have:

\[
\frac{1}{4}>\frac{1}{16} \quad \text { or } \quad\left(\frac{1}{2}\right)^{2}>\left(\frac{1}{4}\right)^{2}
\]

We take the logarithm and obtain:

\[
2 \lg \left(\frac{1}{2}\right)>4 \lg \left(\frac{1}{2}\right)
\]

We divide by \(\lg \left(\frac{1}{2}\right)\) and obtain \(2>4\).

Where is the error?
}

\[
\begin{aligned}
2 \lg \left(\frac{1}{2}\right) & >4 \lg \left(\frac{1}{2}\right) \\
2 \lg (1)-2 \lg (2) & >4 \lg (1)-4 \lg (2) \\
-2 \lg (2)>-4 \lg (2) & \equiv-2>-4 \equiv 2<4
\end{aligned}
\]

The division by \(\lg \left(\frac{1}{2}\right)\) was incorrect in this way. Since \(\lg \left(\frac{1}{2}\right)<0\), the inequality sign should have been reversed when dividing.