<s>[INST] How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$? I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common. 
 [/INST] Since $5=2^{\log _2 5}$ we have
$$5^{19}=2^{19 \log_2 5}.$$
Now we get some estimates on that logarithm:
We have $2^{2.25}=4 \sqrt[4]{2}<5$, as $\sqrt[4]{2} < \frac{5}{4}=1.25$. This means $\log_2 5>2.25$ and thus
$$5^{19}=2^{19 \log_2 5}> 2^{19 \times 2.25}=2^{42.75}>2^{39}.$$
Also,
$$52^7=2^{7 \log_2 52} $$
and since $2^{5 \frac{2}{3}}=\frac{64}{\sqrt[3]{2}}<52$ we have
$$52^7=2^{7 \log_2 52}>2^{7\times 5 \frac{2}{3}}=2^{39 \frac{2}{3}}>2^{39}. $$
In summary, we have
$$5^{19}>52^7>2^{39}. $$
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