$$
\log _b^a=\frac{\log _x^a}{\log _x^b}
$$
Then, \(\log _2^3=\frac{\log _x^3}{\log _x^2}, \log _3^4=\frac{\log _x^4}{\log _x^3} \ldots\)
Then,
\(\log _2^3 \cdot \log _3^4 \cdot \log _4^5 \cdot \log _5^6 \cdot \log _6^7 \cdot \log _7^8\)
\(=\frac{\lg _x^3}{\log _x^2} \times \frac{\operatorname{lgg}_x^4}{\log _x^3} \times \frac{\operatorname{lgg}_x^5}{\log _x^4} \times \frac{\log _x^6}{\log _x^5} \times \frac{\log _x^7}{\log _x^6} \times \frac{\log _x^8}{\log _x^7}\)
\(=\frac{\log _x^8}{\log _x^2}=\log _2^8=\log _2^{2^3}=3 \log _2^2\)
\(=3 \times 1=3\)