Answer: \(p=3\).

Apply the laws of logarithms carefully, paying attention to the base of each logarithm and any restrictions on the argument.

Use \(\frac{1}{\log_pa}=\log_ap\). The equation becomes

\(\log_a(p+1)+\log_ap-\log_a(p+2)+\log_a5=\log_a12.\)

Combining logarithms,

\(\log_a\left(\frac{5p(p+1)}{p+2}\right)=\log_a12.\)

Thus

\(\frac{5p(p+1)}{p+2}=12.\)

So \(5p^2+5p=12p+24\), giving

\(5p^2-7p-24=0.\)

Factorising,

\((5p+8)(p-3)=0.\)

The value \(p=-\frac85\) is not valid for the logarithms in the original equation, so \(p=3\).

This gives the required answer: \(p=3\).