<s>[INST] How to move from powers to simple logarithms? I'm following a book that briefly moves from
$$16000 \times 2^{\displaystyle \left (-\frac{x}{24} \right )} = 1600$$
to
$$x = \frac{24 (\log(2) + \log(5))}{\log(2)}$$
adding the comments that
$$\log(1600) = 6\log(2) + 2\log(5) \\
\log(16000) = 7\log(2) + 3\log(5)$$
What general principles are used to achieve this and how to spot possible ways to apply a similar operation in the future?
 [/INST] In general:
\begin{align*}
\log a^b &= b \log a\\
\log (a \cdot b) &= \log a + \log b
\end{align*}
Specific to the above problem:
$$\log(1600) = \log(2^6 \cdot 5^2) = \log(2^6) + \log (5^2) = 6 \log 2 + 2 \log 5$$
$$\log(16000) = \log(2^7 \cdot 5^3) = 7 \log 2 + 3 \log 5$$
So,
\begin{align*}
16000 \times 2^{-x/24} &= 1600\\
\log 16000 + (-x/24) \log 2 &= \log 1600\\
(-x/24)\log 2 &= -\log 2 - \log 5\\
x &=24 \cdot \frac{\log 2 + \log 5}{\log 2}
\end{align*}
</s>