Logarithmic expression how to simplify $$ \log_{3}24 - 3\log_{3}5\times \log_{5}2$$
What I can get is:
$$ \log_3{24} - \log_3{5^3} \times \log_{5}2$$
Change of base rule to get it all in base 3:
$$ \log_5{2} = \frac{\log_3{2}}{\log_3{5}} $$
Now I have:
$$\log_3{24} - \frac{\log_3{5^3}\times \log_3{2}}{\log_3{5}}$$
How to continue from here?

Continuing from what you have,
\begin{align}
\log_3{24} - \frac{\log_3{5^3}\times \log_3{2}}{\log_3{5}} &= \log_3{24} - \frac{3\cdot\log_3{5}\cdot \log_3{2}}{\log_3{5}} \\
&=(\log_3{3} + \log_3{8}) - 3\cdot \log_3{2} \\
&= 1 +  3 \log_3 2 - 3 \log_3 2 \\
&= 1.
\end{align}
In questions like these, it's often a good idea to reduce exponents as much as possible: you started by converting $3 \log_3 5$ to $\log_3 (5^3)$, but the former expression is simpler to work with. 
You can see this further in how expanding $\log_3 24$ was a useful step to finish.
