G10.2 If $\log _{10}(k-1)-\log _{10}\left(k^{2}-5 k+4\right)+1=0$, find $k$.
$$
\begin{array}{l}
10(k-1)=k^{2}-5 k+4 \\
k^{2}-15 k+14=0 \\
k=1 \text { or } 14
\end{array}
$$

When $k=1$, LHS is undefined $\therefore$ rejected
When $k=14$, LHS $=\log _{10} 13-\log _{10}(14-1)(14-4)+1=$ RHS
$$
\therefore k=14
$$