<s>[INST] Find the integer $m$ such that $(\frac{1}{5})^m \cdot (\frac{1}{4})^ {24} = \frac{1}{2(10)^{47}}$ I have this problem
$$(\frac{1}{5})^m \cdot (\frac{1}{4})^ {24} = \frac{1}{2(10)^{47}}$$
and I need to find the integer $m$.
I know that I can write $5^m \cdot 4^{24} = 2(10)^{47} \Leftrightarrow 5^m = \frac{2(10)^{47}}{(1/4)^{24}}$, but I don't know how to go on from there.
It is a question in a GMAT test, so it should be possible to solve using basic operations (i.e. without logarithmic functions).
 [/INST] $$\frac{1}{2(10)^{47}}=\frac{1}{2}\frac{1}{2^{47}}\frac{1}{5^{47}}=\frac{1}{2^{48}}\frac{1}{5^{47}}=\frac{1}{4^{24}}\frac{1}{5^{47}}=\left(\frac{1}{4}\right)^{24}\left(\frac{1}{5}\right)^{47}$$
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