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. 

Hint: you can establish the right order between $5^{19}$ and $2^{39}$ as follows
\begin{align}
5^{19} &= 5^{20-1}\\
&=\frac{5^{20}}{5}\\
&=\frac{(5^2)^{10}}{5}\\
&=\frac{25^{10}}{5}\\
&>\frac{16^{10}}{2}\\
&=\frac{(2^4)^{10}}{2}\\
&=2^{39}
\end{align}
then note that
\begin{align}
52^7 &= 52^{10-3}\\
&=\frac{52^{10}}{52^3}\\
&=\frac{26^{10}\cdot 2^{10}}{26^{3}\cdot 2^{3}}\\
&<\frac{25^{10}\cdot 2^{10}}{26^{3}\cdot 2^{3}}\\
&<\frac{25^{10}\cdot 2^{10}}{26^{3}\cdot 5}\\
&=\frac{5^{20-1}\cdot 2^{10}}{26^{3}}\\
&=5^{19} \cdot \frac{2^{10}}{26^3}\\
&<5^{19}
\end{align}
and now you have to find if $52^7$ is somewhere in between $5^{19}$ and $2^{39}$ or under $2^{39}$.
