<s>[INST] Proof the logaritmic identity Please help me proving the basic logarithmic identity $\log_3 12=1+\log_5 4\cdot \log_6 5\cdot \log_3 6$
 [/INST] Let $\gcd(a,b) = 1$ and $x = a^nb$ where $a,b,n \in \mathbb{Z}$, then 
$$\begin{aligned}
\log_a x &= n + \frac{\ln b}{\ln a} = n + \frac{\ln b}{\ln(b+1)}\frac{\ln(b+1)}{\ln(b+2)}\cdots \frac{\ln(b+(m-1))}{\ln(b+m)}\frac{\ln (b+m)}{\ln a} \newline 
&= n + \log_{b+1} b\log_{b+2}{b+1}\cdots \log_{b+m} (b + (m-1)) \ln_{a} (b+m).
\end{aligned}$$
Take case that $a = 3$ and $b = 4$, $n = 1$ and $m = 2$, then $\log_3 12 = 1 + \log_5 4 \log_6 5 \log_3 6$.
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