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. 

$$ 52^7 = (26^7 \cdot 2^7) > ((2^{4.7})^7 \cdot 2^7) = 2^{39.9} > 2^{39} $$
As $26 = 2^{\frac{\ln 26}{ \ln 2}} \approx 2^{4.7004}$.
Also,
$$ 52^7 < 6^7 \cdot 10^7 = 6^7 5^7 2^7 < 6^7 5^7 5^4 = 5^{7 \cdot \frac{\ln 6}{\ln 5}} 5^7 5^4 = 5^{7.793 + 7 + 4} < 5^{19}$$
Using $2^7 < 5^4$.
Thus,
$$ 2^{39} < 52^7 < 5^{19}$$
